The monochromatic column problem with a prime number of colors

Abstract

Let $p_1,\dots ,p_n$ be a sequence of $n$ pairwise coprime positive integers, $P=p_1\cdots p_n$, and $0,\dots ,m-1$ be a sequence of $m$ different colors. Let $A$ be an $n\times mP$ matrix of colors in which row $i$ consists of blocks of $p_i$ consecutive entries of the same color with colors 0 through $m-1$ repeated cyclically. The monochromatic column problem is to determine the number of columns of $A$ in which every entry is the same color. The solution for a prime number of colors is provided.