Numerical semigroups of Szemerédi type
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 n...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/135998 |
| Acceso en línea: | https://hdl.handle.net/11441/135998 https://doi.org/10.1016/j.dam.2018.03.023 |
| Access Level: | acceso abierto |
| Palabra clave: | Arithmetic progression van der Waerden number Multiplicity Frobenius numbe Conductor |
| Sumario: | 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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