Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs |
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Authors: | Carl Bürger Louis DeBiasio Hannah Guggiari Max Pitz |
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Affiliation: | 1. University of Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany;2. Miami University, Department of Mathematics, Oxford, OH, 45056, United States;3. University of Oxford, Mathematical Institute, Oxford, OX2 6GG, United Kingdom |
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Abstract: | Let 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 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 -edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length for any colour can be covered by pairwise disjoint monochromatic complete symmetric digraphs in colour .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 is bounded by . |
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Keywords: | Complete symmetric digraph Monochromatic path partition Edge-colourings |
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