Source: wikibot/szemeredi-s-theorem
= Szemerédi's theorem
{wiki=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.