Matchings with few colors in colored complete graphs and hypergraphs

Abstract

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.