Some further results in Ramsey graph construction

Abstract

A construction described by the current author (2017) uses two linear prototypes to build a compound graph with Ramsey properties inherited from the prototype graphs. The resulting graph is linear; and cyclic if both prototypes are cyclic. However, it will not generate a cyclic graph from a general linear prototype. Building on the properties of that construction, this paper proves that a general linear prototype graph of order m can be extended using a single new colour to produce a new cyclic graph of order $3m - 1$ which is triangle-free in the new colour, and has the same clique-number as the prototype in every other colour. The paper then describes a cyclic Ramsey $(3;3;4;4; 173)$-graph derived by constrained tree search, thus proving that $R(3;3;4;4) \ge 174$. Using a quadrupling construction to produce a further cyclic graph, it is shown that $R(3;4;5;5) \ge 693$. A compound cyclic Ramsey $(3;7;7; 622)$-graph derived by a limited manual search is then described. Further construction steps produce a $(8;8;8; 6131)$-graph, showing that $R_3(8) \ge 6132$. The paper concludes by showing that $R_4(7) \ge 81206$ and $R_4(9) \ge 630566$, implying corresponding improvements in the lower bounds for $R_5(7)$ and $R_5(9)$ and beyond. These results follow from the existence of cyclic prototype graphs derived by Mathon-Shearer 'doubling'.