Numerical semigroups of Szemerédi type
Résumé
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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