A note on multicolour Erd\H{o}s-Hajnal conjecture

Abstract

Informally, the Erd\H{o}s-Hajnal conjecture (shortly EH-conjecture) asserts that if a sufficiently large host clique on $n$ vertices is edge-coloured avoiding a copy of some fixed edge-coloured clique, then there is a large homogeneous set of size $n^\beta$ for some positive $\beta$, where a set of vertices is homogeneous if it does not induce all the colours. This conjecture, if true, claims that imposing local conditions on edge-partitions of cliques results in a global structural consequence such as a large homogeneous set, a set avoiding all edges of some part. While this conjecture attracted a lot of attention, it is still open even for two colours. In this note, we reduce the multicolour EH-conjecture to the case when the number of colours used in a host clique is either the same as in the forbidden pattern or one more. We exhibit a non-monotonicity behaviour of homogeneous sets in coloured cliques with forbidden patterns by showing that allowing an extra colour in the host graph could actually decrease the size of a largest homogeneous set.