2018/05/04 by Carl Bürger, Bürger, Carl, Louis DeBiasio +5
Mathematics · #05C15 #05C20 #05C35 #05C63 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C15 #msc:05C20 #msc:05C35 #msc:05C63
paper · pdf · doi:10.48550/arxiv.1805.01699
11 pages. This article supersedes arXiv:1710.10900 and arXiv:1711.08711
arxiv created 2018/05/04 · arxiv updated 2018/05/07
Let Kℕ 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 Kℕ 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](1)/(ℓi).