Almost empty monochromatic triangles in planar point sets

Abstract

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: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.

Empty Rainbow Triangles in $k$-colored Point Sets

“…We say that an r-hole on S is monochromatic if all its vertices are of the same color, and that it is rainbow 1 if all its vertices are of different colors. Many chromatic variants on problems regarding r-holes in colored points sets have been studied since; see [6,23,1,3,7,21,4,19,8,14,25]. In particular, Aichholzer, Fabila-Monroy, Flores-Peñaloza, Hackl, Huemer, and Urrutia showed that every 2-colored set of n points in the plane determines Ω(n 5/4 ) empty monochromatic triangles [1].…”