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

We show that for every η > 0 there exists an integer n 0 such that every 2‐coloring of the 3‐uniform complete hypergraph on n ≥ n 0 vertices contains two disjoint monochromatic tight cycles of distinct colors that together cover all but at most η n vertices. The same result holds if tight cycles are replaced by loose cycles.

Confirming a conjecture of Gyárfás, we prove that, for all natural numbers k and r , the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r , the vertices of every r-edge-coloured complete graph can be partitioned into a bounded number of p-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further extends these results to all host hypergraphs of bounded independence number.

Extending a result of Rado to hypergraphs, we prove that for all s, k, t ∈ N with k ≥ t ≥ 2, the vertices of every r = s(k − t + 1)-edge-coloured countably infinite complete k-graph can be partitioned into the cores of at most s monochromatic t-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.

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