Finding Certain Arithmetic Progressions in 2-Coloured Cyclic Groups

TL;DR

The paper investigates which pairs of integers allow for guaranteed monochromatic arithmetic progressions with specified color patterns in large prime cyclic groups, establishing new findability results and counterexamples.

Contribution

It proves that pairs of the form (2, k) are findable in large prime cyclic groups, extending previous results, and provides counterexamples for (3, k) and (14, 14).

Findings

(2, k) is findable for all k.

(3, 30000) is not findable.

(14, 14) is not findable.

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.