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 δ > 0 there exists a p₀ such that for any prime p ≥ p₀ and any red-blue colouring of ℤ/pℤ in which each colour has density at least δ, we can find an arithmetic progression of length a + b inside ℤ/pℤ whose first a elements are red and whose last b elements are blue.

Szemeredi’s Theorem on arithmetic progressions implies that (0, k) and (1, k) are find-able 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.