Some further results in Ramsey graph construction

TL;DR

This paper advances Ramsey graph construction by proving new bounds and methods for creating cyclic graphs with specific properties, leading to improved lower bounds for various Ramsey numbers.

Contribution

It introduces a generalized construction method for cyclic Ramsey graphs using linear prototypes and demonstrates new lower bounds for multiple Ramsey numbers.

Findings

Proves that linear prototype graphs can be extended to cyclic graphs with specific properties.

Establishes new lower bounds for Ramsey numbers R(3;3;4;4), R(3;4;5;5), R_3(8), R_4(7), and R_4(9).

Describes cyclic Ramsey graphs derived by constrained search and manual methods.

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'.