Matchings with few colors in colored complete graphs and hypergraphs
The t-color Ramsey problem for hypergraph matchings was settled by the well-known result of Alon, Frankl and Lovász (answering a conjecture of Erdős). This result was the last step in a chain of special cases most notably Lovász’s solution to Kneser’s problem. We proposed an extension of the Erdős p...
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Published in | Discrete mathematics Vol. 343; no. 5; p. 111831 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier B.V
01.05.2020
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Subjects | |
Online Access | Get full text |
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Summary: | The t-color Ramsey problem for hypergraph matchings was settled by the well-known result of Alon, Frankl and Lovász (answering a conjecture of Erdős). This result was the last step in a chain of special cases most notably Lovász’s solution to Kneser’s problem. We proposed an extension of the Erdős problem: for given 1≤s≤t, what is the maximum number of vertices that can be covered by a matching having at most s colors in every t-coloring of the edges of the complete graph Kn (or hypergraph Knr).
We revisit the first unknown case, r=2,s=2,t=4, where we conjectured that in every 4-coloring of Kn there is a bicolored matching covering at least ⌊3n∕4⌋ vertices. We prove that this is true asymptotically by applying a recent twist of a standard application of the Regularity method: instead of lifting a (bicolored) matching of the reduced graph to regular cluster pairs, we lift a (bicolored) basic 2-matching, a subgraph whose connected components are edges and odd cycles. To find the bicolored basic 2-matching with at least ⌊3n∕4⌋ vertices in every 4-coloring of Kn we use Tutte’s minimax formula. |
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ISSN: | 0012-365X 1872-681X |
DOI: | 10.1016/j.disc.2020.111831 |