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.
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