Monochromatic tree covers and Ramsey numbers for set-coloured graphs

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 Király et al., see below.) However, we also show that the bound r−k is not best possible in general.