Numerical semigroups of Szemerédi type

Abstract

Given any length k ≥ 3 and density 0 < δ ≤ 1, we introduce and study the set Sz(k, δ) consisting of all positive integers n such that every subset of {1, 2, . . . , n} of density at least δ contains an arithmetic progression of length k. A famous theorem of Szemerédi guarantees that this set is not empty. We show that Sz(k, δ)∪{0} is a numerical semigroup and we determine it for (k, δ) = (4, 1/2) and for more than thirty pairs (3, δ) with δ > 1/5.