Improved Lower Bounds for Multicolour Ramsey Numbers using SAT-Solvers

Abstract: This paper sets out the results of a range of searches for linear and cyclic graph colourings with specific Ramsey properties. The new graphs comprise mainly 'template graphs' which can be used in a construction described by the current author in 2021 to build linear or cyclic compound graphs with inherited Ramsey properties. These graphs result in improved lower bounds for a wide range of multicolour Ramsey numbers. Searches were carried out using relatively simple programs (written in the language C') to generate clauses for input to the PeneLoPe and Plingeling parallel SAT-solvers. When solutions were found, the output from the solvers specified the desired graph colourings. The majority of the graphs produced by this work aretemplate graphs' with parameters in the form $(k,k,3)$ or $(k,l,3)$ with $k \ne l$. Using these template graphs in familiar constructions, it has been possible to demonstrate significant improvements for lower bounds for most $R_r(k)$ for $5 \le k \le 9$ and $r \ge 4$. These improvements provide correspondingly increased lower bounds on $\Gamma(k) = \lim_{\substack{r \rightarrow \infty}} R{_r}(k)^{1/r}$. We also show that $R_3(8) \ge 7174$ and $R_3(9) \ge 15041$. Other new lower bounds include $R(3,6,6) \ge 338$ and $R(3,8,8) \ge 941$, based on non-template cyclic graphs, and the interesting particular cases $R(3,4,5,5) \ge 729$ and $R(3,5,5,5) \ge 1429$. A spreadsheet containing specimens of many of the graphs mentioned here will be attached as an ArXiv ancillary file.