Finding Certain Arithmetic Progressions in 2-Coloured Cyclic Groups

Abstract

We say a pair of integers $(a, b)$ is findable if the following is true. For any $\delta > 0$ there exists a $p_0$ such that for any prime $p \ge p_0$ and any red-blue colouring of $\mathbb{Z} /p\mathbb{Z}$ in which each colour has density at least $\delta$, we can find an arithmetic progression of length $a+b$ inside $\mathbb{Z}/p\mathbb{Z}$ whose first $a$ elements are red and whose last $b$ elements are blue. Szemer\'edi's Theorem on arithmetic progressions implies that $(0,k)$ and $(1,k)$ are findable for any $k$. We prove that $(2, k)$ is also findable for any $k$. However, the same is not true of $(3, k)$. Indeed, we give a construction showing that $(3, 30000)$ is not findable. We also show that $(14, 14)$ is not findable.