Monochromatic tree covers and Ramsey numbers for set-coloured graphs

Abstract: We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-uniform hypergraphs: they have a transversal of size at most $r-k$. (Similar results have been obtained by Kir\'aly et al., see below.) However, we also show that the bound $r-k$ is not best possible in general.