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![Page 1: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/1.jpg)
The minimum number of disjoint pairs in setsystems and related problems
Shagnik Das
University of California, Los Angeles
Aug 7, 2013
Joint work with Wenying Gan and Benny Sudakov
![Page 2: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/2.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems
Definition (Intersecting systems)
A set system F is said to be intersecting if F1 ∩ F2 6= ∅ for allF1,F2 ∈ F .
Observation
If a system F on [n] is intersecting, |F| ≤ 2n−1.
Proof: F can contain at most one of F ,F c for all F ⊂ [n].
Constructions:
Star: all sets containing 1(n odd) All sets of size at least n+1
2
![Page 3: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/3.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems
Definition (Intersecting systems)
A set system F is said to be intersecting if F1 ∩ F2 6= ∅ for allF1,F2 ∈ F .
Observation
If a system F on [n] is intersecting, |F| ≤ 2n−1.
Proof: F can contain at most one of F ,F c for all F ⊂ [n].
Constructions:
Star: all sets containing 1(n odd) All sets of size at least n+1
2
![Page 4: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/4.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems
Definition (Intersecting systems)
A set system F is said to be intersecting if F1 ∩ F2 6= ∅ for allF1,F2 ∈ F .
Observation
If a system F on [n] is intersecting, |F| ≤ 2n−1.
Proof: F can contain at most one of F ,F c for all F ⊂ [n].
Constructions:
Star: all sets containing 1(n odd) All sets of size at least n+1
2
![Page 5: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/5.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems
Definition (Intersecting systems)
A set system F is said to be intersecting if F1 ∩ F2 6= ∅ for allF1,F2 ∈ F .
Observation
If a system F on [n] is intersecting, |F| ≤ 2n−1.
Proof: F can contain at most one of F ,F c for all F ⊂ [n].
Constructions:
Star: all sets containing 1(n odd) All sets of size at least n+1
2
![Page 6: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/6.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems: Erdos-Ko-Rado
Theorem (Erdos-Ko-Rado, 1961)
Suppose n ≥ 2k, and F ⊂([n]k
)is intersecting. Then |F| ≤
(n−1k−1).
Extremal systems: stars
1
A star with centre 1
![Page 7: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/7.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems: Erdos-Ko-Rado
Theorem (Erdos-Ko-Rado, 1961)
Suppose n ≥ 2k, and F ⊂([n]k
)is intersecting. Then |F| ≤
(n−1k−1).
Extremal systems: stars
1
A star with centre 1
![Page 8: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/8.jpg)
Historical Background Our Results Concluding Remarks
Intersecting systems: Erdos-Ko-Rado
Theorem (Erdos-Ko-Rado, 1961)
Suppose n ≥ 2k, and F ⊂([n]k
)is intersecting. Then |F| ≤
(n−1k−1).
Extremal systems: stars
1
A star with centre 1
![Page 9: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/9.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold
Previous results answer the typical extremal problem
Question
How large can a structure be without containing a forbiddenconfiguration?
Gives rise to natural extension
Question
How many forbidden configurations must appear in largerstructures?
![Page 10: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/10.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold
Previous results answer the typical extremal problem
Question
How large can a structure be without containing a forbiddenconfiguration?
Gives rise to natural extension
Question
How many forbidden configurations must appear in largerstructures?
![Page 11: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/11.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: one extra set
Warm-up: How many disjoint pairs must a system of 2n−1 + 1sets contain?
Answer: 1
Any maximal intersecting system can be extended to haveonly one disjoint pair
Let F be an intersecting system with 2n−1 sets, and letF0 ∈ F be minimal
Adding F c0 to F only creates the disjoint pair {F0,F c
0 }
![Page 12: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/12.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: one extra set
Warm-up: How many disjoint pairs must a system of 2n−1 + 1sets contain?
Answer:
1
Any maximal intersecting system can be extended to haveonly one disjoint pair
Let F be an intersecting system with 2n−1 sets, and letF0 ∈ F be minimal
Adding F c0 to F only creates the disjoint pair {F0,F c
0 }
![Page 13: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/13.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: one extra set
Warm-up: How many disjoint pairs must a system of 2n−1 + 1sets contain?
Answer: 1
Any maximal intersecting system can be extended to haveonly one disjoint pair
Let F be an intersecting system with 2n−1 sets, and letF0 ∈ F be minimal
Adding F c0 to F only creates the disjoint pair {F0,F c
0 }
![Page 14: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/14.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: one extra set
Warm-up: How many disjoint pairs must a system of 2n−1 + 1sets contain?
Answer: 1
Any maximal intersecting system can be extended to haveonly one disjoint pair
Let F be an intersecting system with 2n−1 sets, and letF0 ∈ F be minimal
Adding F c0 to F only creates the disjoint pair {F0,F c
0 }
![Page 15: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/15.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: one extra set
Warm-up: How many disjoint pairs must a system of 2n−1 + 1sets contain?
Answer: 1
Any maximal intersecting system can be extended to haveonly one disjoint pair
Let F be an intersecting system with 2n−1 sets, and letF0 ∈ F be minimal
Adding F c0 to F only creates the disjoint pair {F0,F c
0 }
![Page 16: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/16.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 17: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/17.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 18: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/18.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 19: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/19.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 20: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/20.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 21: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/21.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 22: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/22.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 23: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/23.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 24: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/24.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 25: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/25.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 26: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/26.jpg)
Historical Background Our Results Concluding Remarks
Beyond the threshold: many extra sets
Previous argument:( [n]> n
2
)construction best to extend
Theorem (Frankl, 1977; Ahlswede, 1980)
Suppose∣∣∣( [n]>k
)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of
disjoint pairs in a system of m sets is attained by some F with( [n]>k
)⊂ F ⊂
( [n]≥k).
0 1
. . .
n−32
n−12
n+12
n+32
. . .
n − 1 n
![Page 27: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/27.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)
Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2 . . . s
![Page 28: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/28.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2 . . . s
![Page 29: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/29.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1
2 . . . s
![Page 30: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/30.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1
2 . . . s
![Page 31: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/31.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1
2 . . . s
![Page 32: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/32.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2
. . . s
![Page 33: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/33.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2
. . . s
![Page 34: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/34.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2
. . . s
![Page 35: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/35.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2 . . . s
![Page 36: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/36.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2 . . . s
![Page 37: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/37.jpg)
Historical Background Our Results Concluding Remarks
The k-uniform setting
Extremal systems depend on pairs in([n]k
)Question (Ahlswede, 1980)
Which systems in([n]k
)minimise the number of disjoint pairs?
Natural construction: union of stars
1 2 . . . s
![Page 38: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/38.jpg)
Historical Background Our Results Concluding Remarks
A conjecture
Conjecture (Bollobas-Leader, 2000)
For small systems, a union of stars minimises the number ofdisjoint pairs.
A system is optimal iff its complement is
Conjecture ⇒ for large systems, a clique is optimal
Conjecture holds for k = 2
Theorem (Ahlswede-Katona, 1978)
The n-vertex graph with m edges and the minimal number ofdisjoint pairs of edges is:
a union of stars if m < 12
(n2
)− n
2 , and
a clique if m > 12
(n2
)+ n
2 .
![Page 39: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/39.jpg)
Historical Background Our Results Concluding Remarks
A conjecture
Conjecture (Bollobas-Leader, 2000)
For small systems, a union of stars minimises the number ofdisjoint pairs.
A system is optimal iff its complement is
Conjecture ⇒ for large systems, a clique is optimal
Conjecture holds for k = 2
Theorem (Ahlswede-Katona, 1978)
The n-vertex graph with m edges and the minimal number ofdisjoint pairs of edges is:
a union of stars if m < 12
(n2
)− n
2 , and
a clique if m > 12
(n2
)+ n
2 .
![Page 40: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/40.jpg)
Historical Background Our Results Concluding Remarks
A conjecture
Conjecture (Bollobas-Leader, 2000)
For small systems, a union of stars minimises the number ofdisjoint pairs.
A system is optimal iff its complement is
Conjecture ⇒ for large systems, a clique is optimal
Conjecture holds for k = 2
Theorem (Ahlswede-Katona, 1978)
The n-vertex graph with m edges and the minimal number ofdisjoint pairs of edges is:
a union of stars if m < 12
(n2
)− n
2 , and
a clique if m > 12
(n2
)+ n
2 .
![Page 41: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/41.jpg)
Historical Background Our Results Concluding Remarks
A conjecture
Conjecture (Bollobas-Leader, 2000)
For small systems, a union of stars minimises the number ofdisjoint pairs.
A system is optimal iff its complement is
Conjecture ⇒ for large systems, a clique is optimal
Conjecture holds for k = 2
Theorem (Ahlswede-Katona, 1978)
The n-vertex graph with m edges and the minimal number ofdisjoint pairs of edges is:
a union of stars if m < 12
(n2
)− n
2 , and
a clique if m > 12
(n2
)+ n
2 .
![Page 42: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/42.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure
![Page 43: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/43.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure
![Page 44: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/44.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure
![Page 45: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/45.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on s
Step 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure
![Page 46: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/46.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on sStep 2: Existence of a popular element
Step 3: Existence of a cover of size sStep 4: Determining the exact structure
![Page 47: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/47.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size s
Step 4: Determining the exact structure
![Page 48: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/48.jpg)
Historical Background Our Results Concluding Remarks
New result
We verify the Bollobas-Leader conjecture
Theorem (D.-Gan-Sudakov, 2013+)
Given n > 108k2s(k + s), and(nk
)−(n−s+1
k
)≤ m ≤
(nk
)−(n−s
k
),
then the minimum number of disjoint pairs for a system of m setsin([n]k
)is attained by taking s − 1 full stars and a partial star.
Proof outline:
Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure
![Page 49: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/49.jpg)
Historical Background Our Results Concluding Remarks
Notation
dp(F) = # of disjoint pairs in Fdp(F ,G) = # of disjoint pairs between F and G
F(i) = {F ∈ F : i ∈ F}A(i) =
([n]k
)(i) = {F ⊂ [n] : |F | = k , i ∈ F}
X is a cover for F if for every F ∈ F , F ∩ X 6= ∅
![Page 50: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/50.jpg)
Historical Background Our Results Concluding Remarks
Notation
dp(F) = # of disjoint pairs in Fdp(F ,G) = # of disjoint pairs between F and G
F(i) = {F ∈ F : i ∈ F}A(i) =
([n]k
)(i) = {F ⊂ [n] : |F | = k , i ∈ F}
X is a cover for F if for every F ∈ F , F ∩ X 6= ∅
![Page 51: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/51.jpg)
Historical Background Our Results Concluding Remarks
Notation
dp(F) = # of disjoint pairs in Fdp(F ,G) = # of disjoint pairs between F and G
F(i) = {F ∈ F : i ∈ F}A(i) =
([n]k
)(i) = {F ⊂ [n] : |F | = k , i ∈ F}
X is a cover for F if for every F ∈ F , F ∩ X 6= ∅
![Page 52: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/52.jpg)
Historical Background Our Results Concluding Remarks
Step one: induction
Base case: s = 1
Trivial: we have an intersecting system
Induction step: s ≥ 2
If F has a full star, say F(1), then we have
dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))
=
(n − k − 1
k − 1
)(m −
(n − 1
k − 1
))+ dp(F \ F(1)).
Induction ⇒ dp(F \ F(1)) minimised by a union of stars
Hence we may assume there are no full starsGiven F ∈ F and i ∈ [n], can replace F by a set containing i
![Page 53: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/53.jpg)
Historical Background Our Results Concluding Remarks
Step one: induction
Base case: s = 1
Trivial: we have an intersecting system
Induction step: s ≥ 2
If F has a full star, say F(1), then we have
dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))
=
(n − k − 1
k − 1
)(m −
(n − 1
k − 1
))+ dp(F \ F(1)).
Induction ⇒ dp(F \ F(1)) minimised by a union of stars
Hence we may assume there are no full starsGiven F ∈ F and i ∈ [n], can replace F by a set containing i
![Page 54: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/54.jpg)
Historical Background Our Results Concluding Remarks
Step one: induction
Base case: s = 1
Trivial: we have an intersecting system
Induction step: s ≥ 2
If F has a full star, say F(1), then we have
dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))
=
(n − k − 1
k − 1
)(m −
(n − 1
k − 1
))+ dp(F \ F(1)).
Induction ⇒ dp(F \ F(1)) minimised by a union of stars
Hence we may assume there are no full starsGiven F ∈ F and i ∈ [n], can replace F by a set containing i
![Page 55: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/55.jpg)
Historical Background Our Results Concluding Remarks
Step one: induction
Base case: s = 1
Trivial: we have an intersecting system
Induction step: s ≥ 2
If F has a full star, say F(1), then we have
dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))
=
(n − k − 1
k − 1
)(m −
(n − 1
k − 1
))+ dp(F \ F(1)).
Induction ⇒ dp(F \ F(1)) minimised by a union of stars
Hence we may assume there are no full starsGiven F ∈ F and i ∈ [n], can replace F by a set containing i
![Page 56: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/56.jpg)
Historical Background Our Results Concluding Remarks
Step one: induction
Base case: s = 1
Trivial: we have an intersecting system
Induction step: s ≥ 2
If F has a full star, say F(1), then we have
dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))
=
(n − k − 1
k − 1
)(m −
(n − 1
k − 1
))+ dp(F \ F(1)).
Induction ⇒ dp(F \ F(1)) minimised by a union of stars
Hence we may assume there are no full stars
Given F ∈ F and i ∈ [n], can replace F by a set containing i
![Page 57: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/57.jpg)
Historical Background Our Results Concluding Remarks
Step one: induction
Base case: s = 1
Trivial: we have an intersecting system
Induction step: s ≥ 2
If F has a full star, say F(1), then we have
dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))
=
(n − k − 1
k − 1
)(m −
(n − 1
k − 1
))+ dp(F \ F(1)).
Induction ⇒ dp(F \ F(1)) minimised by a union of stars
Hence we may assume there are no full starsGiven F ∈ F and i ∈ [n], can replace F by a set containing i
![Page 58: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/58.jpg)
Historical Background Our Results Concluding Remarks
Step two: popular element
If F is extremal, we must have dp(F) ≤ 12
(1− 1
s
)m2
Goal: |F(i)| ≥ mks for some i (wlog i = 1)
Union-bound:
dp(F) =1
2
∑F∈F
dp(F ,F) ≥ 1
2
∑F∈F
(m −
∑i∈F|F(i)|
)
≥ 1
2
∑F∈F
(m − k |F(1)|) =1
2
(1− k |F(1)|
m
)m2.
Thus k|F(1)|m ≥ 1
s ⇒ |F(1)| ≥ mks
![Page 59: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/59.jpg)
Historical Background Our Results Concluding Remarks
Step two: popular element
If F is extremal, we must have dp(F) ≤ 12
(1− 1
s
)m2
Goal: |F(i)| ≥ mks for some i (wlog i = 1)
Union-bound:
dp(F) =1
2
∑F∈F
dp(F ,F) ≥ 1
2
∑F∈F
(m −
∑i∈F|F(i)|
)
≥ 1
2
∑F∈F
(m − k |F(1)|) =1
2
(1− k |F(1)|
m
)m2.
Thus k|F(1)|m ≥ 1
s ⇒ |F(1)| ≥ mks
![Page 60: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/60.jpg)
Historical Background Our Results Concluding Remarks
Step two: popular element
If F is extremal, we must have dp(F) ≤ 12
(1− 1
s
)m2
Goal: |F(i)| ≥ mks for some i (wlog i = 1)
Union-bound:
dp(F) =1
2
∑F∈F
dp(F ,F) ≥ 1
2
∑F∈F
(m −
∑i∈F|F(i)|
)
≥ 1
2
∑F∈F
(m − k |F(1)|) =1
2
(1− k |F(1)|
m
)m2.
Thus k|F(1)|m ≥ 1
s ⇒ |F(1)| ≥ mks
![Page 61: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/61.jpg)
Historical Background Our Results Concluding Remarks
Step two: popular element
If F is extremal, we must have dp(F) ≤ 12
(1− 1
s
)m2
Goal: |F(i)| ≥ mks for some i (wlog i = 1)
Union-bound:
dp(F) =1
2
∑F∈F
dp(F ,F) ≥ 1
2
∑F∈F
(m −
∑i∈F|F(i)|
)
≥ 1
2
∑F∈F
(m − k |F(1)|) =1
2
(1− k |F(1)|
m
)m2.
Thus k|F(1)|m ≥ 1
s ⇒ |F(1)| ≥ mks
![Page 62: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/62.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover
We have dp(F ,F) = m − |∪i∈FF(i)|⇒ m −
∑i∈F |F(i)| ≤ dp(F ,F) ≤ m −maxi∈F |F(i)|
Since we can shift sets to F(1), we must have, for all F ∈ F ,∑i∈F |F(i)| ≥ |F(1)|
⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover
X is small:
km =∑
F∈F |F | =∑
i |F(i)| ≥∑
i∈X |F(i)| ≥ 1k |F(1)| |X |
⇒ |X | ≤ km|F(1)| ≤ k3s.
![Page 63: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/63.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover
We have dp(F ,F) = m − |∪i∈FF(i)|⇒ m −
∑i∈F |F(i)| ≤ dp(F ,F) ≤ m −maxi∈F |F(i)|
Since we can shift sets to F(1), we must have, for all F ∈ F ,∑i∈F |F(i)| ≥ |F(1)|
⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover
X is small:
km =∑
F∈F |F | =∑
i |F(i)| ≥∑
i∈X |F(i)| ≥ 1k |F(1)| |X |
⇒ |X | ≤ km|F(1)| ≤ k3s.
![Page 64: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/64.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover
We have dp(F ,F) = m − |∪i∈FF(i)|⇒ m −
∑i∈F |F(i)| ≤ dp(F ,F) ≤ m −maxi∈F |F(i)|
Since we can shift sets to F(1), we must have, for all F ∈ F ,∑i∈F |F(i)| ≥ |F(1)|
⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover
X is small:
km =∑
F∈F |F | =∑
i |F(i)| ≥∑
i∈X |F(i)| ≥ 1k |F(1)| |X |
⇒ |X | ≤ km|F(1)| ≤ k3s.
![Page 65: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/65.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover
We have dp(F ,F) = m − |∪i∈FF(i)|⇒ m −
∑i∈F |F(i)| ≤ dp(F ,F) ≤ m −maxi∈F |F(i)|
Since we can shift sets to F(1), we must have, for all F ∈ F ,∑i∈F |F(i)| ≥ |F(1)|
⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover
X is small:
km =∑
F∈F |F | =∑
i |F(i)| ≥∑
i∈X |F(i)| ≥ 1k |F(1)| |X |
⇒ |X | ≤ km|F(1)| ≤ k3s.
![Page 66: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/66.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover
We have dp(F ,F) = m − |∪i∈FF(i)|⇒ m −
∑i∈F |F(i)| ≤ dp(F ,F) ≤ m −maxi∈F |F(i)|
Since we can shift sets to F(1), we must have, for all F ∈ F ,∑i∈F |F(i)| ≥ |F(1)|
⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover
X is small:
km =∑
F∈F |F | =∑
i |F(i)| ≥∑
i∈X |F(i)| ≥ 1k |F(1)| |X |
⇒ |X | ≤ km|F(1)| ≤ k3s.
![Page 67: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/67.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 68: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/68.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 69: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/69.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 70: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/70.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 71: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/71.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 72: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/72.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 73: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/73.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 74: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/74.jpg)
Historical Background Our Results Concluding Remarks
Step three: small cover (II)
Almost all intersections take place in X
X
F
Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets
⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |
⇒ dp(F) = 12
(1− 1
|X | + o(1))
m2.
Since dp(F) ≤ 12
(1− 1
s
)m2, we have |X | = s
![Page 75: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/75.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersectingdp(G,A) =
∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for
all G ∈ G
�
![Page 76: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/76.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersectingdp(G,A) =
∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for
all G ∈ G
�
![Page 77: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/77.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersectingdp(G,A) =
∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for
all G ∈ G
�
![Page 78: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/78.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersectingdp(G,A) =
∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for
all G ∈ G
�
![Page 79: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/79.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersecting
dp(G,A) =∑
G∈G dp(G ,A) maximised when |G ∩ X | = 1 forall G ∈ G
�
![Page 80: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/80.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersectingdp(G,A) =
∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for
all G ∈ G
�
![Page 81: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/81.jpg)
Historical Background Our Results Concluding Remarks
Step four: exact structure
We may now assume [s] is a cover for F
Let A = ∪si=1A(i) be all sets meeting [s].
Then F ⊂ A; let G = A \ F .
Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)
Minimised when F consists of s − 1 full stars and one partialstar
dp(G) = 0 as G is intersectingdp(G,A) =
∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for
all G ∈ G
�
![Page 82: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/82.jpg)
Historical Background Our Results Concluding Remarks
Further results
Characterisation of all extremal systems
t-disjoint pairs:
We say F1,F2 are t-disjoint if |F1 ∩ F2| < tUsing similar methods, we determine which small systemsminimise the number of t-disjoint pairsWhen t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory
Can also minimise the number of q-matchings
![Page 83: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/83.jpg)
Historical Background Our Results Concluding Remarks
Further results
Characterisation of all extremal systems
t-disjoint pairs:
We say F1,F2 are t-disjoint if |F1 ∩ F2| < t
Using similar methods, we determine which small systemsminimise the number of t-disjoint pairsWhen t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory
Can also minimise the number of q-matchings
![Page 84: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/84.jpg)
Historical Background Our Results Concluding Remarks
Further results
Characterisation of all extremal systems
t-disjoint pairs:
We say F1,F2 are t-disjoint if |F1 ∩ F2| < tUsing similar methods, we determine which small systemsminimise the number of t-disjoint pairs
When t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory
Can also minimise the number of q-matchings
![Page 85: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/85.jpg)
Historical Background Our Results Concluding Remarks
Further results
Characterisation of all extremal systems
t-disjoint pairs:
We say F1,F2 are t-disjoint if |F1 ∩ F2| < tUsing similar methods, we determine which small systemsminimise the number of t-disjoint pairsWhen t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory
Can also minimise the number of q-matchings
![Page 86: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/86.jpg)
Historical Background Our Results Concluding Remarks
Further results
Characterisation of all extremal systems
t-disjoint pairs:
We say F1,F2 are t-disjoint if |F1 ∩ F2| < tUsing similar methods, we determine which small systemsminimise the number of t-disjoint pairsWhen t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory
Can also minimise the number of q-matchings
![Page 87: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/87.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 88: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/88.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systems
What are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 89: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/89.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?
The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 90: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/90.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 91: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/91.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 92: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/92.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersecting
Have previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 93: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/93.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)
More complicated relationship in the k-uniform setting
![Page 94: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/94.jpg)
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions
Most probably intersecting (Katona-Katona-Katona):
Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
![Page 95: The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe minimum number of disjoint pairs in set systems and related problems Shagnik Das University](https://reader033.vdocuments.site/reader033/viewer/2022042115/5e92a3a3c3b99174e36178de/html5/thumbnails/95.jpg)
Historical Background Our Results Concluding Remarks