Problems on the Triangular Lattice

In this work, we consider a number of problems defined on the triangular lattice with n rows, which we will denote as Tₙ. Define a proper coloring to be an assignment of colors to the points of Tₙ 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 Tₙ. We either determine exactly or give upper bounds for $f(n)$ for many small values of n, and it is shown that limn→∞ f(n)/n ≤ 13. We also give formulas counting the number of pairs of points in Tₙ for which there are, respectively, 0, 1, or 2 choices of points in Tₙ which extend those two into the vertices of an equilateral triangle. Along the way, we pose a number of related questions.