bichromatic and equichromatic lines in c 2 and r 2 george b purdy justin w smith 1

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Bichromatic and Equichromatic Lines in C2 and R2

George B PurdyJustin W Smith

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Problem Definition• Definition

• The “Orchard”

Problem

• Graham’s Question

• Fukuda’s Conjecture

• Pach & Pinchasi’s

Results

• Kleitman & Pinchasi’s

Conjecture

• Our Results

• Conclusion

A bichromatic set of points.

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Problem




Results


Conjecture

• Our Results

• Conclusion

A bichromatic line.

4



Problem




Results


Conjecture

• Our Results

• Conclusion

A monochromatic line.

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• A Problem of

Sylvester




Results


Conjecture

• Our Results

• Conclusion

A Few Interesting questions.

• How many bichromatic lines must there be? (I.e., a “lower bound”.)

• How many bichromatic lines pass through at most 4 points?

• How many bichromatic lines pass through at most 6 points?

• Must there always exist a monochromatic line? (Any two points “determine” a line. We will only consider determined lines.)

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The “Orchard” Problem• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

• The “Orchard Problem” appeared in books in the early 19th century.

• Below is from Rational Amusements For Winter Evenings (1821), a book containing Geometric and Arithmetic puzzles.

• The questions asks how to plant 9 trees such that there are 10 rows of three.

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The “Orchard” Problem• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

How many three point lines?

• Consider the following configuration of trees (represented by points).

• How many “3-tree rows” are there?

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Some Notation• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Eight three-point lines.

• Let tk= # of lines passing through exactly k points.

• In this configuration t3=8.

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Counting• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Eight three-point lines..

• What value is t2?

• We assume tk = 0 for all k > 3.• In this example these answers are obvious, but

is there a general rule we can apply to assist this counting?

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Counting• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Eight three-point lines.

• Given n points:

• Since t3=8, we know t2=12, since

3(8) + 1(12) = = 36

222

nt

kk

k

2

9

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History• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Ten three-point lines.

• In this configuration, t3=10.• So how many two-point lines?

3(10)+1(6)= = 36Thus, t2=6.

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9

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Problem




Results


Conjecture

• Our Results

• Conclusion

Sylvester-Gallai Theorem.

• Can t2 = 0?• One of the earliest results in the field is

called the Sylvester-Gallai Theorem: Every finite set of noncollinear points in a

plane determines a two-point line (a.k.a. an “ordinary” line).

• This question was asked by Erdős around 1933, and remained open until solved by Gallai around 1944.

• It was later discovered that Sylvester had asked the same question in 1893 (but no solution was given).

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Problem




Results


Conjecture

• Our Results

• Conclusion

Melchior’s Inequality.

• Melchior’s Inequality:

• Thus, t2 ≥ 3 + t4 + 2t5 + 4t6 +…• Was published in 1940 (even before

Gallai’s proof).• Melchior’s Inequality is derived from

Euler’s Polyhedral Formula.• Open Problem: It is conjectured that t2 ≥ n/2.

Best known result is t2 ≥ 6n/13

3)3(2

k

ktk

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Problem




Results


Conjecture

• Our Results

• Conclusion

Graham’s Question.

• Ron Graham asked around 1965 whether every bichromatic configuration of lines determines a monochromatic point.

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Problem




Results


Conjecture

• Our Results

• Conclusion

Graham’s Question.

• Motzkin-Rabin Theorem: Any bichromatic set of non-concurrent lines determines a monochromatic intersection point.

• By “Principle of Duality” the theorem also makes an equivalent statement concerning bichromatic sets of points.

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Recent History• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Fukuda’s Conjecture.

• In 1996, Fukuda proposed a generalization of the Sylvester-Gallai Theorem.

• Let R and B be two sets of points. • If:– R and B are separated by a straight line.– |R| and |B| differ by at most one.Then there exists a bichromatic ordinary

line (i.e., a two-point line).• Now known to be false for a specific

configuration of 9 points.

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Problem




Results


Conjecture

• Our Results

• Conclusion

Motivation for Pach and Pinchasi.

• Fukuda’s Conjecture provided motivation for Pach and Pinchasi to demonstrate that bichromatic lines must exist with “relatively few” points, even without the separating line restriction.

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Problem




Results


Conjecture

• Our Results

• Conclusion

Pach and Pinchasi’s result.

• The primary result of Pach and Pinchasi’s paper (from 2000) is the following theorem:

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Problem




Results


Conjecture

• Our Results

• Conclusion

Kleitman and Pinchasi’s conjecture.

• In 2003, Kleitman and Pinchasi had the following conjecture:Let G be a set of n red points and n or n-1 blue points. If no color class is contained in a line, then G determines at least |G|-1 bichromatic lines.

• They were able to prove |G|-3 using Linear Programming.

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Important Observations• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Pach and Pinchasi’s observation.

• Let ti,j be the number of lines passing through exactly i red points and j blue points. Assume, |R| = |B|= n

• Pach & Pinchasi had the following two observations.

- Bichromatic pairs:

- Monochromatic pairs:

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Problem




Results


Conjecture

• Our Results

• Conclusion

Pach and Pinchasi’s observation.

Bichromatic pairs:

Monochromatic pairs:

(# Bichromatic) – (# Monochromatic):

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Problem




Results


Conjecture

• Our Results

• Conclusion

Our observation.

• Assume, |R|=n and |B|= n-k• These formulas become the following:- Bichromatic pairs:

- Monochromatic pairs:

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Problem




Results


Conjecture

• Our Results

• Conclusion

Our observation.

• We also subtracted the two equations. But we did one additional step which allowed us to achieve better results.

(# Bichromatic) – (# Monochromatic):

• The squared coefficient on the right side effectively limits the number of monochromatic lines.

• Equichromatic lines are those passing through i red points and j blue points where |i-j|≤1.

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Hirzebruch’s Inequalities• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Application to C2.

• In 1983, Hirzebruch published an inequality very similar to Melchior’s.

• This result was derived from Algebraic Geometry, and thus applies to the complex plane (and also to the real plane).

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Our Results• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Bichromatic Equichromatic Lines.

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Problem




Results


Conjecture

• Our Results

• Conclusion

Proved Kleitman-Pinchasi Conjecture.

• We proved the Kleitman-Pinchasi Conjecture for n≥79.

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Problem




Results


Conjecture

• Our Results

• Conclusion

Application to C2.

• We proved an analogue of the Kelly-Moser Theorem for the Complex Plane.

• This allowed us to also prove a “Kleitman-Pinchasi Conjecture” for the Complex Plane.

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Conclusion• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Many Open Problems Remain.

• Our paper, “Bichromatic and Equichromatic Lines in C2 and R2,” is to appear in Discrete and Computational Geometry.

• There are still many open problems related to this research:– The Motzkin-Dirac Conjecture (i.e., n/2

ordinary lines).– Maximal configurations for the Orchard

Problem on > 12 points.– Must there exist t/2 bichromatic lines?

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Questions?• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

Questions?

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References• Definition


Problem




Results


Conjecture

• Our Results

• Conclusion

References

• J. Pach, and R. Pinchasi, Bichromatic Lines with Few Points. Journal of Combinatorial Theory, Series A 90 (2000) 326-335.

• D.J. Kleitman, and R. Pinchasi, A Note on the Number of Bichromatic Lines, http://www-math.mit.edu/~room/ps_files/KP_bichnum.ps, Massachusetts Institute of Technology, 2003.

• F. Hirzebruch, Singularities of Algebraic Surfaces and Characteristic Numbers. in: D. Sundararaman, S. Gitler, and A. Verjovsky, (Eds.), The Lefschetz Centennial Conference, American Mathematical Society, Providence, R.I., 1984, pp. 141-155.

• J. Jackson, Rational Amusement for Winter Evenings. Bristol: Longman, Hurst, Rees, Orme and Brown, 1821.

bichromatic and equichromatic lines in c 2 and r 2 george b purdy justin w smith 1

Documents

problem definition definition

results conclusion

orchard problem definition

counting definition

history definition

bichromatic lines

point lines

notation definition