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K á ra-P ó r-Wood conjecture Big Line or Big Clique in Planar Point Sets

K á ra-P ó r-Wood conjecture Big Line or Big Clique in Planar Point Sets. Jozef Jirasek jirasekjozef@gmail.com. The Problem. Given two integers k , l Show that if we have “ enough ” points in the plane, then there are either:. enough. There exists an integer N( k , l ),

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K á ra-P ó r-Wood conjecture Big Line or Big Clique in Planar Point Sets

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  1. Kára-Pór-Wood conjectureBig Line or Big Cliquein Planar Point Sets Jozef Jirasek jirasekjozef@gmail.com

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

  3. Simple cases • 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!

  4. 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!

  5. 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 thek – 1 points

  6. 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!

  7. 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? jirasekjozef@gmail.com

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