"Almost empty monochromatic triangles in planar point sets" by Deepan Basu, Kinjal Basu et al.
 

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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