Monochromatic tree covers and Ramsey numbers for set-coloured graphs

We extend results on monochromatic tree covers and from classical Ramsey theory to a generalised setting, where each of the edges of an underlying host graph (here, either a complete graph or a complete bipartite graph), is coloured with a set of colours. Our results for tree covers in this setting have an application to Ryser’s Conjecture. Every r-partite r-uniform hypergraph whose edges pairwise intersect in at least k ≥ (r − 2)/2 vertices, has a transversal of size at most r − k. In particular, for these hypergraphs, Ryser’s conjecture holds.