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- W2801096345 abstract "Let $K_{mathbb{N}}$ 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 $K_{mathbb{N}}$ 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 $ell_i$ for any colour $ileq r$ can be covered by $prod_{ileq r} ell_i$ 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 $prod_{iin [r]}frac{1}{ell_i}$." @default.
- W2801096345 created "2018-05-17" @default.
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- W2801096345 date "2018-05-04" @default.
- W2801096345 modified "2023-09-27" @default.
- W2801096345 title "Partitioning Edge-Coloured Complete Symmetric Digraphs into Monochromatic Complete Subgraphs" @default.
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