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Journal Articles Discrete Applied Mathematics Year : 2019

Numerical semigroups of Szemerédi type

S.D. Adhikari
  • Function : Author
L. Boza
M.P. Revuelta
M.I. Sanz
  • Function : Author

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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Dates and versions

hal-04147368 , version 1 (30-06-2023)

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S.D. Adhikari, L. Boza, Shalom Eliahou, M.P. Revuelta, M.I. Sanz. Numerical semigroups of Szemerédi type. Discrete Applied Mathematics, 2019, 263, pp.8-13. ⟨10.1016/j.dam.2018.03.023⟩. ⟨hal-04147368⟩
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