New lower bounds for Schur and weak Schur numbers

Romain Ageron (1) , Paul Casteras (1) , Thibaut Pellerin (1) , Yann Portella (1) , Arpad Rimmel (2) , Joanna Tomasik (2, 3)

1 CentraleSupélec
2 LISN - Laboratoire Interdisciplinaire des Sciences du Numérique
3 GALaC - Graphes, Algorithmes et Combinatoire - LISN

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]

Romain Ageron, Paul Casteras, Thibaut Pellerin, Yann Portella, Arpad Rimmel, et al.. New lower bounds for Schur and weak Schur numbers. 2022. ⟨hal-04377719⟩