Szemerédi's theorem

Szemerédi's theorem is a fundamental result in combinatorial number theory which pertains to arithmetic progressions in sets of integers. Specifically, the theorem states that for any positive integer \( k \), any subset of the integers with positive density contains a non-trivial arithmetic progression of length \( k \). More formally, if \( A \) is a subset of the positive integers with positive upper density, i.e.