Almost empty monochromatic triangles in planar point sets

For positive integers c,s≥1c,s≥1, let M3(c,s)M3(c,s) be the least integer such that any set of at least M3(c,s)M3(c,s) points in the plane, no three on a line and colored with cc colors, contains a monochromatic triangle with at most ss interior points. The case s=0s=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)=3M3(1,0)=3, M3(2,0)=9M3(2,0)=9 and M3(c,0)=∞M3(c,0)=∞, for c≥3c≥3. In this paper we extend these results when c≥2c≥2 and s≥1s≥1. We prove that the least integer λ3(c)λ3(c) such that M3(c,λ3(c))<∞M3(c,λ3(c))<∞ satisfies: ⌊c−12⌋≤λ3(c)≤c−2, where c≥2c≥2. Moreover, the exact values of M3(c,s)M3(c,s) are determined for small values of cc and ss. We also conjecture that λ3(4)=1λ3(4)=1, and verify it for sufficiently large Horton sets.