Bichromatic and Equichromatic Lines in C 2 and R 2 George B Purdy Justin W Smith 1

Problem Definition Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 2 A bichromatic set of points.

Problem Definition Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 3 A bichromatic line.

Problem Definition Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 4 A monochromatic line.

Problem Definition Definition A Problem of Sylvester Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 5 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.)

The “Orchard” Problem Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 6 The “Orchard Problem” appeared in books in the early 19 th 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.

The “Orchard” Problem Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 7 How many three point lines? Consider the following configuration of trees (represented by points). How many “3-tree rows” are there?

Some Notation Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 8 Eight three-point lines. Let t k = # of lines passing through exactly k points. In this configuration t 3 =8.

Counting Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 9 Eight three-point lines.. What value is t 2 ? We assume t k = 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?

Counting Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 10 Eight three-point lines. Given n points: Since t 3 =8, we know t 2 =12, since 3(8) + 1(12) = = 36

History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 11 Ten three-point lines. In this configuration, t 3 =10. So how many two-point lines? 3(10)+1(6)= = 36 Thus, t 2 =6.

History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 12 Sylvester-Gallai Theorem. Can t 2 = 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 It was later discovered that Sylvester had asked the same question in 1893 (but no solution was given).

History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 13 Melchior’s Inequality. Melchior’s Inequality: Thus, t 2 ≥ 3 + t 4 + 2t 5 + 4t 6 +… 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 t 2 ≥ n/2. Best known result is t 2 ≥ 6n/13

History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 14 Graham’s Question. Ron Graham asked around 1965 whether every bichromatic configuration of lines determines a monochromatic point.

History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 15 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.

Recent History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 16 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.

Recent History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 17 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.

Recent History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 18 Pach and Pinchasi’s result. The primary result of Pach and Pinchasi’s paper (from 2000) is the following theorem:

Recent History Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 19 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.

Important Observations Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 20 Pach and Pinchasi’s observation. Let t i,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:

Important Observations Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 21 Pach and Pinchasi’s observation. Bichromatic pairs: Monochromatic pairs: (# Bichromatic) – (# Monochromatic):

Important Observations Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 22 Our observation. Assume, |R|=n and |B|= n-k These formulas become the following: -Bichromatic pairs: - Monochromatic pairs:

Important Observations Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 23 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.

Hirzebruch’s Inequalities Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 24 Application to C 2. 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).

Our Results Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 25 Bichromatic Equichromatic Lines.

Our Results Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 26 Proved Kleitman-Pinchasi Conjecture. We proved the Kleitman-Pinchasi Conjecture for n≥79.

Our Results Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 27 Application to C 2. 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.

Conclusion Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 28 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?

Questions? Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 29 Questions?

References Definition The “Orchard” Problem Graham’s Question Fukuda’s Conjecture Pach & Pinchasi’s Results Kleitman & Pinchasi’s Conjecture Our Results Conclusion 30 References J. Pach, and R. Pinchasi, Bichromatic Lines with Few Points. Journal of Combinatorial Theory, Series A 90 (2000) D.J. Kleitman, and R. Pinchasi, A Note on the Number of Bichromatic Lines, math.mit.edu/~room/ps_files/KP_bichnum.ps, Massachusetts Institute of Technology, 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 J. Jackson, Rational Amusement for Winter Evenings. Bristol: Longman, Hurst, Rees, Orme and Brown, 1821.