A generalised linear Ramsey graph construction

Abstract

A construction described by the current author in 2017 uses two linear `prototype' graphs to build a compound graph with Ramsey properties inherited from the prototypes. This paper describes a generalisation of that construction which has produced improved lower bounds in many cases for multicolour Ramsey numbers. The resulting graphs are linear, as before, and under certain specific conditions set out here, they can be cyclic. The mechanism of the new construction requires that the first prototype contains a triangle-free `template' in one colour, with defined properties. This paper shows that in the compound graph, clique numbers in the colours of the first prototype may exceed those of the prototype. However, it proves necessary only to test for a limited subset of the possible cliques using these colours, in order to evaluate the relevant clique numbers for the entire graph. Clique numbers in the colours of the second prototype are equal to those of that prototype. These attributes enable the efficient searching of a new family of graphs based on the `template' approach. It has also been found that there are a number of useful cases in which the clique numbers of the first prototype are not increased by the compounding process. As a result of this construction many lower bounds can be improved. The improvements include $R_4(5) \ge 4073$, $R_5(5) \ge 38914$, $R_3(6) \ge 1106$, $R_4(6) \ge 21302$, $R_4(7) \ge 84623$ and $R_3(9) \ge 14034$. It is also shown that $R_3(9) \ge 14081$ using a non-linear construction.