Kára-Pór-Wood conjecture Big Line or Big Clique in Planar Point Sets Jozef Jirasek

The Problem Given two integers k, l Show that if we have “enough” points in the plane, then there are either: k points which can “see” each other l points which all lie on a single line or There exists an integer N( k, l ), such that for any n ≥ N, every arrangement of n points contains either: enough

l = 3(three points on a line) –set N = k, from n ≥ N points pick any k. –if they can see each other, we are done. –if two of them can not see each other, –we get a line with 3 points! Simple cases

k = 3(3 points which see each other) –set N = l, let n ≥ N. –if all n points lie on a line, we are done. –otherwise, pick the smallest triangle. if two points can not see each other, the triangle was not the smallest! –therefore, three points of the smallest triangle must be able to see each other!

Proof by Induction? N(3, l ) = l (pick the smallest triangle) For larger k : Select N( k – 1, l ) points Find either: –l points on a line, or –k – 1 points “seeing” each other Find another point which “sees” all the k – 1 points

Will not work! Given k – 1 points which see each other, we can add an arbitrary number of points, such that: – no l points lie on a single line, and – no added point sees all the k – 1 points!

Known results Easy for k ≤ 3 or l ≤ 3 (as shown here). Kára, Pór, Wood: k ≤ 4, all l. Addario-Berry et al.: k = 5, l = 4. Abel et al.: k = 5, all l. Questions? Ideas?