New lower bounds for Schur and weak Schur numbers
Résumé
This article provides new lower bounds for both Schur and weak Schur numbers by exploiting a "template"-based approach. The concept of "template" is also generalized to weak Schur numbers. Finding new templates leads to explicit partitions improving lower bounds as well as the growth rate for Schur numbers, weak Schur numbers, and multicolor Ramsey numbers $R_n(3)$. The new lower bounds include $S(9) \geq 17\,803$, $S(10) \geq 60\,948$, $\mathit{WS}(6) \geq 646$, $\mathit{WS}(9) \geq 22\,536$ and $\mathit{WS}(10) \geq 71\,256$.
Mots clés
Schur number Weak Schur number Ramsey theory Sum-free partition MSC 2020: 05D10 05A17 11B75 11P81
Schur number
Weak Schur number
Ramsey theory
Schur number Weak Schur number Ramsey theory Sum-free partition MSC 2020: 05D10 05A17 11B75 11P81
Sum-free partition MSC 2020: 05D10
05A17
11B75
11P81
Combinatorics (math.CO)
FOS: Mathematics
05D10
Domaines
Mathématique discrète [cs.DM]| Origine | Fichiers produits par l'(les) auteur(s) |
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