Problems on the Triangular Lattice

TL;DR

This paper investigates proper colorings of a triangular lattice to avoid monochromatic equilateral triangles, providing bounds on the minimum number of colors needed and counting specific point configurations.

Contribution

It determines or bounds the minimum colors for proper lattice colorings and explores related combinatorial configurations and questions.

Findings

Exact or bounded values of f(n) for small n

Limit of f(n)/n as n approaches infinity is at most 1/3

Formulas for counting point pairs with specific equilateral triangle extensions

Abstract

In this work, we consider a number of problems defined on the triangular lattice with $n$ rows, which we will denote as $T_n$. Define a \textit{proper coloring} to be an assignment of colors to the points of $T_n$ such that no three points constituting the vertices of an equilateral triangle all receive the same color, and denote by $f(n)$ the smallest possible number of colors that can be used in a proper coloring of $T_n$. We either determine exactly or give upper bounds for $f(n)$ for many small values of $n$, and it is shown that $\lim_{n\to\infty} \frac{f(n)}{n} \leq \frac13$. We also give formulas counting the number of pairs of points in $T_n$ for which there are, respectively, 0, 1, or 2 choices of points in $T_n$ which extend those two into the vertices of an equilateral triangle. Along the way, we pose a number of related questions.