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 are `template 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.