New lower bounds for Schur and weak Schur numbers

2021/12/06 by Paul Casteras, Ageron, Romain, Casteras, Paul +6 · 2 citations
Mathematics · #05A17 #05D10 #11B75 #11P81 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2112.03175

openalex publication_date 2021/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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) ≥ 17 803, S(10) ≥ 60 948, WS(6) ≥ 646, WS(9) ≥ 22 536 and WS(10) ≥ 71 256.

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• The chromatic number of finite projective spaces
• An improved upper bound for the multicolour Ramsey number of odd cycles

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