Almost Empty Monochromatic Triangles in Planar Point Sets

TL;DR

This paper investigates the minimum number of points needed in a colored planar set to guarantee a monochromatic triangle with limited interior points, extending known results to broader parameters and determining exact values for small cases.

Contribution

It extends the understanding of monochromatic triangles with interior point constraints for multiple colors and provides bounds and exact values for specific cases.

Findings

Bounds on (c) for monochromatic triangles with interior points

Exact values of M_3(c, s) for small c and s

Conjecture and verification of (4) for large Horton sets

Abstract

For positive integers $c, s \geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ 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 $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\infty$, for $c\geq 3$. In this paper we extend these results when $c \geq 2$ and $s \geq 1$. We prove that the least integer $\lambda_3(c)$ such that $M_3(c, \lambda_3(c))< \infty$ satisfies: $$\left\lfloor\frac{c-1}{2}\right\rfloor \leq\lambda_3(c)\leq c-2,$$ where $c \geq 2$. Moreover, the exact values of $M_3(c, s)$ are determined for small values of $c$ and $s$. We also conjecture that $\lambda_3(4)=1$, and verify it for sufficiently large Horton sets.