Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

Abstract

Let K→N be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of K→N in which every monochromatic path has density 0.However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every (r+1)-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length ℓi for any colour i≤r can be covered by ∏i∈[r]ℓi pairwise disjoint monochromatic complete symmetric digraphs in colour r+1.Furthermore, we present a stability version for the countable case of the latter result: We prove that the edge-colouring is uniquely determined on a large subgraph, as soon as the upper density of monochromatic paths in colour r+1 is bounded by ∏i∈[r]1ℓi.