ramsey numbers of ordered graphs
TRANSCRIPT
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Ramsey numbers of ordered graphs
Martin Balko, Josef Cibulka, Karel Kral, and Jan Kyncl
Charles University in Prague,Czech Republic
February 27, 2016
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Ramsey theory
“Every sufficiently large system contains a well-organized subsystem.”
Ramsey’s theorem for graphs
For every collection G1, . . . ,Gc of graphs there is a sufficiently largeN = N(G1, . . . ,Gc) such that every c-coloring of the edges of KN contains acopy of Gi in color i for some i ∈ [c].
Ramsey number R(G1, . . . ,Gc) of G1, . . . ,Gc is the smallest such N .
If all G1, . . . ,Gc are isomorphic to G , we write R(G ; c) or R(G ) if c = 2.
Classical bounds of Erdos and Szekeres: 2n/2 ≤ R(Kn) ≤ 22n.
Example: R(K3) = R(C4) = 6
![Page 3: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/3.jpg)
Ramsey theory
“Every sufficiently large system contains a well-organized subsystem.”
Ramsey’s theorem for graphs
For every collection G1, . . . ,Gc of graphs there is a sufficiently largeN = N(G1, . . . ,Gc) such that every c-coloring of the edges of KN contains acopy of Gi in color i for some i ∈ [c].
Ramsey number R(G1, . . . ,Gc) of G1, . . . ,Gc is the smallest such N .
If all G1, . . . ,Gc are isomorphic to G , we write R(G ; c) or R(G ) if c = 2.
Classical bounds of Erdos and Szekeres: 2n/2 ≤ R(Kn) ≤ 22n.
Example: R(K3) = R(C4) = 6
![Page 4: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/4.jpg)
Ramsey theory
“Every sufficiently large system contains a well-organized subsystem.”
Ramsey’s theorem for graphs
For every collection G1, . . . ,Gc of graphs there is a sufficiently largeN = N(G1, . . . ,Gc) such that every c-coloring of the edges of KN contains acopy of Gi in color i for some i ∈ [c].
Ramsey number R(G1, . . . ,Gc) of G1, . . . ,Gc is the smallest such N .
If all G1, . . . ,Gc are isomorphic to G , we write R(G ; c) or R(G ) if c = 2.
Classical bounds of Erdos and Szekeres: 2n/2 ≤ R(Kn) ≤ 22n.
Example: R(K3) = R(C4) = 6
![Page 5: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/5.jpg)
Ramsey theory
“Every sufficiently large system contains a well-organized subsystem.”
Ramsey’s theorem for graphs
For every collection G1, . . . ,Gc of graphs there is a sufficiently largeN = N(G1, . . . ,Gc) such that every c-coloring of the edges of KN contains acopy of Gi in color i for some i ∈ [c].
Ramsey number R(G1, . . . ,Gc) of G1, . . . ,Gc is the smallest such N .
If all G1, . . . ,Gc are isomorphic to G , we write R(G ; c) or R(G ) if c = 2.
Classical bounds of Erdos and Szekeres: 2n/2 ≤ R(Kn) ≤ 22n.
Example: R(K3) = R(C4) = 6
![Page 6: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/6.jpg)
Ramsey theory
“Every sufficiently large system contains a well-organized subsystem.”
Ramsey’s theorem for graphs
For every collection G1, . . . ,Gc of graphs there is a sufficiently largeN = N(G1, . . . ,Gc) such that every c-coloring of the edges of KN contains acopy of Gi in color i for some i ∈ [c].
Ramsey number R(G1, . . . ,Gc) of G1, . . . ,Gc is the smallest such N .
If all G1, . . . ,Gc are isomorphic to G , we write R(G ; c) or R(G ) if c = 2.
Classical bounds of Erdos and Szekeres: 2n/2 ≤ R(Kn) ≤ 22n.
Example: R(K3) = R(C4) = 6
![Page 7: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/7.jpg)
Ramsey theory
“Every sufficiently large system contains a well-organized subsystem.”
Ramsey’s theorem for graphs
For every collection G1, . . . ,Gc of graphs there is a sufficiently largeN = N(G1, . . . ,Gc) such that every c-coloring of the edges of KN contains acopy of Gi in color i for some i ∈ [c].
Ramsey number R(G1, . . . ,Gc) of G1, . . . ,Gc is the smallest such N .
If all G1, . . . ,Gc are isomorphic to G , we write R(G ; c) or R(G ) if c = 2.
Classical bounds of Erdos and Szekeres: 2n/2 ≤ R(Kn) ≤ 22n.
Example: R(K3) = R(C4) = 6
![Page 8: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/8.jpg)
Ordered graphs
An ordered graph G is a pair (G ,≺) where G is a graph and ≺ is a totalordering of its vertices.(H ,≺1) is an ordered subgraph of (G ,≺2) if H ⊆ G and ≺1⊆≺2.The ordered Ramsey number R(G1, . . . ,Gc) for ordered graphsG1, . . . ,Gc is the least number N such that every c-coloring of edges ofKN contains Gi of color i for some i ∈ [c] as an ordered subgraph.
Observation
For ordered graphs G1 = (G1,≺1), . . . ,Gc = (Gc ≺c) we have
R(G1, . . . ,Gc) ≤ R(G1, . . . ,Gc) ≤ R(K|V (G1)|, . . . ,K|V (Gc )|).
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Ordered graphs
An ordered graph G is a pair (G ,≺) where G is a graph and ≺ is a totalordering of its vertices.
(H ,≺1) is an ordered subgraph of (G ,≺2) if H ⊆ G and ≺1⊆≺2.The ordered Ramsey number R(G1, . . . ,Gc) for ordered graphsG1, . . . ,Gc is the least number N such that every c-coloring of edges ofKN contains Gi of color i for some i ∈ [c] as an ordered subgraph.
Observation
For ordered graphs G1 = (G1,≺1), . . . ,Gc = (Gc ≺c) we have
R(G1, . . . ,Gc) ≤ R(G1, . . . ,Gc) ≤ R(K|V (G1)|, . . . ,K|V (Gc )|).
![Page 10: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/10.jpg)
Ordered graphs
An ordered graph G is a pair (G ,≺) where G is a graph and ≺ is a totalordering of its vertices.(H ,≺1) is an ordered subgraph of (G ,≺2) if H ⊆ G and ≺1⊆≺2.
The ordered Ramsey number R(G1, . . . ,Gc) for ordered graphsG1, . . . ,Gc is the least number N such that every c-coloring of edges ofKN contains Gi of color i for some i ∈ [c] as an ordered subgraph.
Observation
For ordered graphs G1 = (G1,≺1), . . . ,Gc = (Gc ≺c) we have
R(G1, . . . ,Gc) ≤ R(G1, . . . ,Gc) ≤ R(K|V (G1)|, . . . ,K|V (Gc )|).
![Page 11: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/11.jpg)
Ordered graphs
An ordered graph G is a pair (G ,≺) where G is a graph and ≺ is a totalordering of its vertices.(H ,≺1) is an ordered subgraph of (G ,≺2) if H ⊆ G and ≺1⊆≺2.The ordered Ramsey number R(G1, . . . ,Gc) for ordered graphsG1, . . . ,Gc is the least number N such that every c-coloring of edges ofKN contains Gi of color i for some i ∈ [c] as an ordered subgraph.
Observation
For ordered graphs G1 = (G1,≺1), . . . ,Gc = (Gc ≺c) we have
R(G1, . . . ,Gc) ≤ R(G1, . . . ,Gc) ≤ R(K|V (G1)|, . . . ,K|V (Gc )|).
![Page 12: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/12.jpg)
Ordered graphs
An ordered graph G is a pair (G ,≺) where G is a graph and ≺ is a totalordering of its vertices.(H ,≺1) is an ordered subgraph of (G ,≺2) if H ⊆ G and ≺1⊆≺2.The ordered Ramsey number R(G1, . . . ,Gc) for ordered graphsG1, . . . ,Gc is the least number N such that every c-coloring of edges ofKN contains Gi of color i for some i ∈ [c] as an ordered subgraph.
Observation
For ordered graphs G1 = (G1,≺1), . . . ,Gc = (Gc ≺c) we have
R(G1, . . . ,Gc) ≤ R(G1, . . . ,Gc) ≤ R(K|V (G1)|, . . . ,K|V (Gc )|).
![Page 13: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/13.jpg)
Ordered graphs
An ordered graph G is a pair (G ,≺) where G is a graph and ≺ is a totalordering of its vertices.(H ,≺1) is an ordered subgraph of (G ,≺2) if H ⊆ G and ≺1⊆≺2.The ordered Ramsey number R(G1, . . . ,Gc) for ordered graphsG1, . . . ,Gc is the least number N such that every c-coloring of edges ofKN contains Gi of color i for some i ∈ [c] as an ordered subgraph.
Observation
For ordered graphs G1 = (G1,≺1), . . . ,Gc = (Gc ≺c) we have
R(G1, . . . ,Gc) ≤ R(G1, . . . ,Gc) ≤ R(K|V (G1)|, . . . ,K|V (Gc )|).
Example:
R(CA) = 10 R(CB) = 11 R(CC ) = 14
CA CB CC
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Known results
The k-uniform monotone path (Pkn ,≺mon) is a k-uniform hypergraph
with n vertices and edges formed by k-tuples of consecutive vertices.
th(x) is a tower function given by t1(x) = x and th(x) = 2th−1(x).
Choudum and Ponnusamy, 2002:R((Pn1 ,≺mon), . . . , (Pnc ,≺mon)) = 1 +
∏ci=1(ni − 1).
Fox, Pach, Sudakov, and Suk, 2011:tk−1(Cnc−1) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(C ′nc−1 log n).
Moshkovitz and Shapira, 2012:tk−1(nc−1/2
√c) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(2nc−1).
Cibulka, Gao, Krcal, Valla, and Valtr, 2013:Every ordered path Pn satisfies R(Pn) ≤ O(nlog n).
Similar results discovered independently by Conlon, Fox, Lee, andSudakov, 2014+.
![Page 15: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/15.jpg)
Known results
The k-uniform monotone path (Pkn ,≺mon) is a k-uniform hypergraph
with n vertices and edges formed by k-tuples of consecutive vertices.
th(x) is a tower function given by t1(x) = x and th(x) = 2th−1(x).
Choudum and Ponnusamy, 2002:R((Pn1 ,≺mon), . . . , (Pnc ,≺mon)) = 1 +
∏ci=1(ni − 1).
Fox, Pach, Sudakov, and Suk, 2011:tk−1(Cnc−1) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(C ′nc−1 log n).
Moshkovitz and Shapira, 2012:tk−1(nc−1/2
√c) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(2nc−1).
Cibulka, Gao, Krcal, Valla, and Valtr, 2013:Every ordered path Pn satisfies R(Pn) ≤ O(nlog n).
Similar results discovered independently by Conlon, Fox, Lee, andSudakov, 2014+.
![Page 16: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/16.jpg)
Known results
The k-uniform monotone path (Pkn ,≺mon) is a k-uniform hypergraph
with n vertices and edges formed by k-tuples of consecutive vertices.
th(x) is a tower function given by t1(x) = x and th(x) = 2th−1(x).
Choudum and Ponnusamy, 2002:R((Pn1 ,≺mon), . . . , (Pnc ,≺mon)) = 1 +
∏ci=1(ni − 1).
Fox, Pach, Sudakov, and Suk, 2011:tk−1(Cnc−1) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(C ′nc−1 log n).
Moshkovitz and Shapira, 2012:tk−1(nc−1/2
√c) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(2nc−1).
Cibulka, Gao, Krcal, Valla, and Valtr, 2013:Every ordered path Pn satisfies R(Pn) ≤ O(nlog n).
Similar results discovered independently by Conlon, Fox, Lee, andSudakov, 2014+.
![Page 17: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/17.jpg)
Known results
The k-uniform monotone path (Pkn ,≺mon) is a k-uniform hypergraph
with n vertices and edges formed by k-tuples of consecutive vertices.
th(x) is a tower function given by t1(x) = x and th(x) = 2th−1(x).
Choudum and Ponnusamy, 2002:R((Pn1 ,≺mon), . . . , (Pnc ,≺mon)) = 1 +
∏ci=1(ni − 1).
Fox, Pach, Sudakov, and Suk, 2011:tk−1(Cnc−1) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(C ′nc−1 log n).
Moshkovitz and Shapira, 2012:tk−1(nc−1/2
√c) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(2nc−1).
Cibulka, Gao, Krcal, Valla, and Valtr, 2013:Every ordered path Pn satisfies R(Pn) ≤ O(nlog n).
Similar results discovered independently by Conlon, Fox, Lee, andSudakov, 2014+.
![Page 18: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/18.jpg)
Known results
The k-uniform monotone path (Pkn ,≺mon) is a k-uniform hypergraph
with n vertices and edges formed by k-tuples of consecutive vertices.
th(x) is a tower function given by t1(x) = x and th(x) = 2th−1(x).
Choudum and Ponnusamy, 2002:R((Pn1 ,≺mon), . . . , (Pnc ,≺mon)) = 1 +
∏ci=1(ni − 1).
Fox, Pach, Sudakov, and Suk, 2011:tk−1(Cnc−1) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(C ′nc−1 log n).
Moshkovitz and Shapira, 2012:tk−1(nc−1/2
√c) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(2nc−1).
Cibulka, Gao, Krcal, Valla, and Valtr, 2013:Every ordered path Pn satisfies R(Pn) ≤ O(nlog n).
Similar results discovered independently by Conlon, Fox, Lee, andSudakov, 2014+.
![Page 19: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/19.jpg)
Known results
The k-uniform monotone path (Pkn ,≺mon) is a k-uniform hypergraph
with n vertices and edges formed by k-tuples of consecutive vertices.
th(x) is a tower function given by t1(x) = x and th(x) = 2th−1(x).
Choudum and Ponnusamy, 2002:R((Pn1 ,≺mon), . . . , (Pnc ,≺mon)) = 1 +
∏ci=1(ni − 1).
Fox, Pach, Sudakov, and Suk, 2011:tk−1(Cnc−1) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(C ′nc−1 log n).
Moshkovitz and Shapira, 2012:tk−1(nc−1/2
√c) ≤ R((Pk
n ,≺mon); c) ≤ tk−1(2nc−1).
Cibulka, Gao, Krcal, Valla, and Valtr, 2013:Every ordered path Pn satisfies R(Pn) ≤ O(nlog n).
Similar results discovered independently by Conlon, Fox, Lee, andSudakov, 2014+.
![Page 20: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/20.jpg)
Specific orderings: ordered stars I
Unordered case (Burr and Roberts, 1973):
R(K1,n−1; c) =
c(n − 2) + 1 if c ≡ n − 1 ≡ 0 (mod 2),
c(n − 2) + 2 otherwise.
Possible orderings:
Sr,s
︸ ︷︷ ︸ ︸ ︷︷ ︸r − 1 s− 1
The 2-colored ordered case was resolved by Choudum and Ponnusamy.
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Specific orderings: ordered stars I
Unordered case (Burr and Roberts, 1973):
R(K1,n−1; c) =
c(n − 2) + 1 if c ≡ n − 1 ≡ 0 (mod 2),
c(n − 2) + 2 otherwise.
Possible orderings:
Sr,s
︸ ︷︷ ︸ ︸ ︷︷ ︸r − 1 s− 1
The 2-colored ordered case was resolved by Choudum and Ponnusamy.
![Page 22: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/22.jpg)
Specific orderings: ordered stars I
Unordered case (Burr and Roberts, 1973):
R(K1,n−1; c) =
c(n − 2) + 1 if c ≡ n − 1 ≡ 0 (mod 2),
c(n − 2) + 2 otherwise.
Possible orderings:
Sr,s
︸ ︷︷ ︸ ︸ ︷︷ ︸r − 1 s− 1
The 2-colored ordered case was resolved by Choudum and Ponnusamy.
![Page 23: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/23.jpg)
Specific orderings: ordered stars I
Unordered case (Burr and Roberts, 1973):
R(K1,n−1; c) =
c(n − 2) + 1 if c ≡ n − 1 ≡ 0 (mod 2),
c(n − 2) + 2 otherwise.
Possible orderings:
Sr,s
︸ ︷︷ ︸ ︸ ︷︷ ︸r − 1 s− 1
The 2-colored ordered case was resolved by Choudum and Ponnusamy.
![Page 24: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/24.jpg)
Specific orderings: ordered stars I
Unordered case (Burr and Roberts, 1973):
R(K1,n−1; c) =
c(n − 2) + 1 if c ≡ n − 1 ≡ 0 (mod 2),
c(n − 2) + 2 otherwise.
Possible orderings:
Sr,s
︸ ︷︷ ︸ ︸ ︷︷ ︸r − 1 s− 1
The 2-colored ordered case was resolved by Choudum and Ponnusamy.
![Page 25: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/25.jpg)
Specific orderings: ordered stars II
Theorem (Choudum and Ponnusamy, 2002)
For positive integers r1, r2, we have R(S1,r1 ,S1,r2) = r1 + r2 − 2 and forr2 ≥ r1 > 2
R(S1,r1 ,Sr2,1) =
⌊−1 +
√1 + 8(r1 − 2)(r2 − 2)
2
⌋+ r1 + r2 − 2.
For arbitrary ordered stars we have
R(S1,r1 ,Sr2,s2) = R(S1,r1 ,Sr2,1) + r1 + s2 − 3
andR(Sr1,s1 ,Sr2,s2) = R(Sr1,1,Sr2,s2) + R(S1,s1 ,Sr2,s2)− 1.
For the multicolored case the ordered Ramsey numbers remain linear inthe number of vertices.
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Specific orderings: ordered stars II
Theorem (Choudum and Ponnusamy, 2002)
For positive integers r1, r2, we have R(S1,r1 ,S1,r2) = r1 + r2 − 2 and forr2 ≥ r1 > 2
R(S1,r1 ,Sr2,1) =
⌊−1 +
√1 + 8(r1 − 2)(r2 − 2)
2
⌋+ r1 + r2 − 2.
For arbitrary ordered stars we have
R(S1,r1 ,Sr2,s2) = R(S1,r1 ,Sr2,1) + r1 + s2 − 3
andR(Sr1,s1 ,Sr2,s2) = R(Sr1,1,Sr2,s2) + R(S1,s1 ,Sr2,s2)− 1.
For the multicolored case the ordered Ramsey numbers remain linear inthe number of vertices.
![Page 27: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/27.jpg)
Specific orderings: ordered stars II
Theorem (Choudum and Ponnusamy, 2002)
For positive integers r1, r2, we have R(S1,r1 ,S1,r2) = r1 + r2 − 2 and forr2 ≥ r1 > 2
R(S1,r1 ,Sr2,1) =
⌊−1 +
√1 + 8(r1 − 2)(r2 − 2)
2
⌋+ r1 + r2 − 2.
For arbitrary ordered stars we have
R(S1,r1 ,Sr2,s2) = R(S1,r1 ,Sr2,1) + r1 + s2 − 3
andR(Sr1,s1 ,Sr2,s2) = R(Sr1,1,Sr2,s2) + R(S1,s1 ,Sr2,s2)− 1.
For the multicolored case the ordered Ramsey numbers remain linear inthe number of vertices.
![Page 28: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/28.jpg)
Specific orderings: ordered cycles
Unordered case (Faudree and Schelp, 1974):
R(Cr ,Cs) =
2r − 1 if (r , s) 6= (3, 3) and 3 ≤ s ≤ r , s is odd,
r + s/2− 1 if (r , s) 6= (4, 4), 4 ≤ s ≤ r , r and s even,
maxr + s/2− 1, 2s − 1 if 4 ≤ s < r , s even, r odd.
A monotone cycle (Cn,≺mon):
Theorem
For integers r ≥ 2 and s ≥ 2, we have
R((Cr ,≺mon), (Cs ,≺mon)) = 2rs − 3r − 3s + 6.
Settles a question of Karolyi et al. about geometric Ramsey numbers.
![Page 29: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/29.jpg)
Specific orderings: ordered cycles
Unordered case (Faudree and Schelp, 1974):
R(Cr ,Cs) =
2r − 1 if (r , s) 6= (3, 3) and 3 ≤ s ≤ r , s is odd,
r + s/2− 1 if (r , s) 6= (4, 4), 4 ≤ s ≤ r , r and s even,
maxr + s/2− 1, 2s − 1 if 4 ≤ s < r , s even, r odd.
A monotone cycle (Cn,≺mon):
Theorem
For integers r ≥ 2 and s ≥ 2, we have
R((Cr ,≺mon), (Cs ,≺mon)) = 2rs − 3r − 3s + 6.
Settles a question of Karolyi et al. about geometric Ramsey numbers.
![Page 30: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/30.jpg)
Specific orderings: ordered cycles
Unordered case (Faudree and Schelp, 1974):
R(Cr ,Cs) =
2r − 1 if (r , s) 6= (3, 3) and 3 ≤ s ≤ r , s is odd,
r + s/2− 1 if (r , s) 6= (4, 4), 4 ≤ s ≤ r , r and s even,
maxr + s/2− 1, 2s − 1 if 4 ≤ s < r , s even, r odd.
A monotone cycle (Cn,≺mon):
Theorem
For integers r ≥ 2 and s ≥ 2, we have
R((Cr ,≺mon), (Cs ,≺mon)) = 2rs − 3r − 3s + 6.
Settles a question of Karolyi et al. about geometric Ramsey numbers.
![Page 31: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/31.jpg)
Specific orderings: ordered cycles
Unordered case (Faudree and Schelp, 1974):
R(Cr ,Cs) =
2r − 1 if (r , s) 6= (3, 3) and 3 ≤ s ≤ r , s is odd,
r + s/2− 1 if (r , s) 6= (4, 4), 4 ≤ s ≤ r , r and s even,
maxr + s/2− 1, 2s − 1 if 4 ≤ s < r , s even, r odd.
A monotone cycle (Cn,≺mon):
Theorem
For integers r ≥ 2 and s ≥ 2, we have
R((Cr ,≺mon), (Cs ,≺mon)) = 2rs − 3r − 3s + 6.
Settles a question of Karolyi et al. about geometric Ramsey numbers.
![Page 32: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/32.jpg)
Specific orderings: ordered cycles
Unordered case (Faudree and Schelp, 1974):
R(Cr ,Cs) =
2r − 1 if (r , s) 6= (3, 3) and 3 ≤ s ≤ r , s is odd,
r + s/2− 1 if (r , s) 6= (4, 4), 4 ≤ s ≤ r , r and s even,
maxr + s/2− 1, 2s − 1 if 4 ≤ s < r , s even, r odd.
A monotone cycle (Cn,≺mon):
Theorem
For integers r ≥ 2 and s ≥ 2, we have
R((Cr ,≺mon), (Cs ,≺mon)) = 2rs − 3r − 3s + 6.
Settles a question of Karolyi et al. about geometric Ramsey numbers.
![Page 33: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/33.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 34: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/34.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 35: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/35.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 36: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/36.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 37: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/37.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 38: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/38.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 39: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/39.jpg)
Specific orderings: ordered paths
Unordered case (Gerenser and Gyarfas, 1967): R(Pr ,Ps) = s − 1 +⌊r2
⌋.
We know that R((Pn,≺mon)) = (n − 1)2 + 1.
The alternating path (Pn,≺alt):
Θ(n) Θ(n2)(Pn, ≺alt)
(Pn, ≺mon)
Proposition
For every positive integer n > 2, we have
2.5n − O(1) ≤ R((Pn,≺alt)) ≤ (2 +√
2)n.
Asymptotically different Ramsey numbers for different orderings.
![Page 40: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/40.jpg)
Bounded-degree ordered graphs
How fast can Ramsey numbers grow for bounded-degree ordered graphs?
Theorem (Chvatal, Rodl, Szemeredi, and Trotter, 1983)
For every ∆ ∈ N there exists C = C (∆) such that for every graph G with nvertices and maximum degree ∆ satisfies
R(G ) ≤ C · n.
Does not hold for ordered graphs.
Theorem
There are arbitrarily large ordered matchings Mn on n vertices such that
R(Mn) ≥ nlog n
5 log log n .
Conlon et al.: almost every ordered n-vertex matching Mn satisfies
R(Mn) ≥ nΩ( log nlog log n).
![Page 41: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/41.jpg)
Bounded-degree ordered graphs
How fast can Ramsey numbers grow for bounded-degree ordered graphs?
Theorem (Chvatal, Rodl, Szemeredi, and Trotter, 1983)
For every ∆ ∈ N there exists C = C (∆) such that for every graph G with nvertices and maximum degree ∆ satisfies
R(G ) ≤ C · n.
Does not hold for ordered graphs.
Theorem
There are arbitrarily large ordered matchings Mn on n vertices such that
R(Mn) ≥ nlog n
5 log log n .
Conlon et al.: almost every ordered n-vertex matching Mn satisfies
R(Mn) ≥ nΩ( log nlog log n).
![Page 42: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/42.jpg)
Bounded-degree ordered graphs
How fast can Ramsey numbers grow for bounded-degree ordered graphs?
Theorem (Chvatal, Rodl, Szemeredi, and Trotter, 1983)
For every ∆ ∈ N there exists C = C (∆) such that for every graph G with nvertices and maximum degree ∆ satisfies
R(G ) ≤ C · n.
Does not hold for ordered graphs.
Theorem
There are arbitrarily large ordered matchings Mn on n vertices such that
R(Mn) ≥ nlog n
5 log log n .
Conlon et al.: almost every ordered n-vertex matching Mn satisfies
R(Mn) ≥ nΩ( log nlog log n).
![Page 43: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/43.jpg)
Bounded-degree ordered graphs
How fast can Ramsey numbers grow for bounded-degree ordered graphs?
Theorem (Chvatal, Rodl, Szemeredi, and Trotter, 1983)
For every ∆ ∈ N there exists C = C (∆) such that for every graph G with nvertices and maximum degree ∆ satisfies
R(G ) ≤ C · n.
Does not hold for ordered graphs.
Theorem
There are arbitrarily large ordered matchings Mn on n vertices such that
R(Mn) ≥ nlog n
5 log log n .
Conlon et al.: almost every ordered n-vertex matching Mn satisfies
R(Mn) ≥ nΩ( log nlog log n).
![Page 44: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/44.jpg)
Bounded-degree ordered graphs
How fast can Ramsey numbers grow for bounded-degree ordered graphs?
Theorem (Chvatal, Rodl, Szemeredi, and Trotter, 1983)
For every ∆ ∈ N there exists C = C (∆) such that for every graph G with nvertices and maximum degree ∆ satisfies
R(G ) ≤ C · n.
Does not hold for ordered graphs.
Theorem
There are arbitrarily large ordered matchings Mn on n vertices such that
R(Mn) ≥ nlog n
5 log log n .
Conlon et al.: almost every ordered n-vertex matching Mn satisfies
R(Mn) ≥ nΩ( log nlog log n).
![Page 45: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/45.jpg)
Bounded-degree ordered graphs
How fast can Ramsey numbers grow for bounded-degree ordered graphs?
Theorem (Chvatal, Rodl, Szemeredi, and Trotter, 1983)
For every ∆ ∈ N there exists C = C (∆) such that for every graph G with nvertices and maximum degree ∆ satisfies
R(G ) ≤ C · n.
Does not hold for ordered graphs.
Theorem
There are arbitrarily large ordered matchings Mn on n vertices such that
R(Mn) ≥ nlog n
5 log log n .
Conlon et al.: almost every ordered n-vertex matching Mn satisfies
R(Mn) ≥ nΩ( log nlog log n).
![Page 46: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/46.jpg)
Growth rate for bounded-degree ordered graphs
The coloring is not constructive.
Corollary
There is arbitrarily large n-vertex graph G with two orderings G ′ and G ′ suchthat R(G) is super-polynomial in n and R(G ′) is linear in n.
![Page 47: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/47.jpg)
Growth rate for bounded-degree ordered graphs
M
The coloring is not constructive.
Corollary
There is arbitrarily large n-vertex graph G with two orderings G ′ and G ′ suchthat R(G) is super-polynomial in n and R(G ′) is linear in n.
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Growth rate for bounded-degree ordered graphs
M
Mk+1
Mk Mk
The coloring is not constructive.
Corollary
There is arbitrarily large n-vertex graph G with two orderings G ′ and G ′ suchthat R(G) is super-polynomial in n and R(G ′) is linear in n.
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Growth rate for bounded-degree ordered graphs
M
Mk+1
Mk Mk
The coloring is not constructive.
Corollary
There is arbitrarily large n-vertex graph G with two orderings G ′ and G ′ suchthat R(G) is super-polynomial in n and R(G ′) is linear in n.
![Page 50: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/50.jpg)
Growth rate for bounded-degree ordered graphs
M
Mk+1
Mk Mk
The coloring is not constructive.
Corollary
There is arbitrarily large n-vertex graph G with two orderings G ′ and G ′ suchthat R(G) is super-polynomial in n and R(G ′) is linear in n.
![Page 51: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/51.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 52: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/52.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 53: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/53.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
χ≺alt(Pn) = 2
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 54: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/54.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
χ≺mon(Pn) = n
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 55: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/55.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
χ≺mon(Pn) = n
Proposition
Every ordered matching (M ,≺) with χ≺(M) = 2 and n vertices satisfies
R((M ,≺)) ≤ O(n2).
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 56: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/56.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
nr︸ ︷︷ ︸ ︸ ︷︷ ︸n︸ ︷︷ ︸
Proposition
Every ordered matching (M ,≺) with χ≺(M) = 2 and n vertices satisfies
R((M ,≺)) ≤ O(n2).
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 57: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/57.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Proposition
Every ordered matching (M ,≺) with χ≺(M) = 2 and n vertices satisfies
R((M ,≺)) ≤ O(n2).
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 58: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/58.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Proposition
Every ordered matching (M ,≺) with χ≺(M) = 2 and n vertices satisfies
R((M ,≺)) ≤ O(n2).
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 59: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/59.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Proposition
Every ordered matching (M ,≺) with χ≺(M) = 2 and n vertices satisfies
R((M ,≺)) ≤ O(n2).
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 60: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/60.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Theorem
For arbitrary k and p every k-degenerate ordered graph (G ,≺) with nvertices and χ≺(G ) = p satisfies
R((G ,≺)) ≤ nO(k)log p
.
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 61: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/61.jpg)
Small ordered Ramsey numbers I
The interval chromatic number χ≺(G ) of (G ,≺) is the minimumnumber of intervals V (G ) can be partitioned into so that no twoadjacent vertices are in the same interval.
Theorem
For arbitrary k and p every k-degenerate ordered graph (G ,≺) with nvertices and χ≺(G ) = p satisfies
R((G ,≺)) ≤ nO(k)log p
.
Conlon et al. showed R(G) ≤ nO(k log p).
![Page 62: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/62.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 63: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/63.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 64: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/64.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 65: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/65.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Bandwidth is 1.
Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 66: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/66.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Bandwidth is n − 1.Bandwidth is 1.
Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 67: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/67.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Bandwidth is n − 1.Bandwidth is 1.Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 68: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/68.jpg)
Small ordered Ramsey numbers II
The length of an edge uv in G = (G ,≺) is |i − j | if u is the ith vertexand v is the jth vertex of G in ≺.
The bandwidth of G is the length of the longest edge in G.
Bandwidth is n − 1.Bandwidth is 1.Theorem
For every k ∈ N, there is a constant C ′k such that every n-vertex orderedgraph G of bandwidth k satisfies R(G) ≤ C ′k · n128k .
Solves a problem of Conlon et al.
![Page 69: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/69.jpg)
Open problems
Specific classes of ordered graphs
Computing precise formulas for other classes of ordered graphs.Multicolored stars, monotone cycles, etc.
Growth rate of ordered Ramsey numbers
Lower bounds for bounded-degree ordered graphs of constantinterval chromatic number?Lower bounds for bounded-degree ordered graphs of constantbandwidth?What is the ordering of a path with minimum ordered Ramseynumber?
Thank you.
![Page 70: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/70.jpg)
Open problems
Specific classes of ordered graphs
Computing precise formulas for other classes of ordered graphs.Multicolored stars, monotone cycles, etc.
Growth rate of ordered Ramsey numbers
Lower bounds for bounded-degree ordered graphs of constantinterval chromatic number?Lower bounds for bounded-degree ordered graphs of constantbandwidth?What is the ordering of a path with minimum ordered Ramseynumber?
Thank you.
![Page 71: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/71.jpg)
Open problems
Specific classes of ordered graphs
Computing precise formulas for other classes of ordered graphs.Multicolored stars, monotone cycles, etc.
Growth rate of ordered Ramsey numbers
Lower bounds for bounded-degree ordered graphs of constantinterval chromatic number?Lower bounds for bounded-degree ordered graphs of constantbandwidth?What is the ordering of a path with minimum ordered Ramseynumber?
Thank you.
![Page 72: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/72.jpg)
Open problems
Specific classes of ordered graphs
Computing precise formulas for other classes of ordered graphs.Multicolored stars, monotone cycles, etc.
Growth rate of ordered Ramsey numbers
Lower bounds for bounded-degree ordered graphs of constantinterval chromatic number?Lower bounds for bounded-degree ordered graphs of constantbandwidth?What is the ordering of a path with minimum ordered Ramseynumber?
Thank you.
![Page 73: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/73.jpg)
Open problems
Specific classes of ordered graphs
Computing precise formulas for other classes of ordered graphs.Multicolored stars, monotone cycles, etc.
Growth rate of ordered Ramsey numbers
Lower bounds for bounded-degree ordered graphs of constantinterval chromatic number?Lower bounds for bounded-degree ordered graphs of constantbandwidth?What is the ordering of a path with minimum ordered Ramseynumber?
Thank you.
![Page 74: Ramsey numbers of ordered graphs](https://reader033.vdocuments.site/reader033/viewer/2022061610/586a5c041a28abf52b8bf855/html5/thumbnails/74.jpg)
Open problems
Specific classes of ordered graphs
Computing precise formulas for other classes of ordered graphs.Multicolored stars, monotone cycles, etc.
Growth rate of ordered Ramsey numbers
Lower bounds for bounded-degree ordered graphs of constantinterval chromatic number?Lower bounds for bounded-degree ordered graphs of constantbandwidth?What is the ordering of a path with minimum ordered Ramseynumber?
Thank you.