Almost empty monochromatic triangles in planar point sets

Document Type

Conference Article

Publication Title

Discrete Applied Mathematics

Abstract

For positive integers c,s≥1, let M3(c,s) be the least integer such that any set of at least M3(c,s) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s=0, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3(1,0)=3, M3(2,0)=9 and M3(c,0)=, for c≥3. In this paper we extend these results when c≥2 and s≥1. We prove that the least integer λ3(c) such that M3(c,λ3(c))< satisfies: c-12≤λ3(c)≤c-2, where c≥2. Moreover, the exact values of M3(c,s) are determined for small values of c and s. We also conjecture that λ3(4)=1, and verify it for sufficiently large Horton sets.

First Page

207

Last Page

213

DOI

10.1016/j.dam.2015.05.033

Publication Date

9-10-2016

Comments

Open Access; Bronze Open Access

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