Ramsey theory constructions from hypergraph matchings

TL;DR

This paper develops asymptotically optimal hypergraph constructions for generalized Ramsey problems, providing new colorings that satisfy specific cycle and clique coloring conditions, answering longstanding questions in combinatorics.

Contribution

It introduces novel hypergraph matching-based constructions for Ramsey theory, offering simplified solutions to classical problems and improving bounds on edge-colorings of complete graphs and bipartite graphs.

Findings

Edge-coloring of bipartite graphs with ~2n/3 colors ensuring 4-cycle diversity.

Edge-coloring of complete graphs with ~5n/6 colors ensuring 4-clique diversity.

Provides asymptotically optimal constructions solving open problems.

Abstract

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle receives at least three colors on its edges. This answers a question of Axenovich, F\"uredi and the second author (On generalized Ramsey theory: the bipartite case, J. Combin. Theory Ser B 79 (2000), 66--86). We also exhibit an edge-coloring of $K_n$ with $5n/6+o(n)$ colors that assigns each copy of $K_4$ at least five colors. This gives an alternative very short solution to an old question of Erd\H{o}s and Gy\'arf\'as that was recently answered by Bennett, Cushman, Dudek, and Pralat by analyzing a colored modification of the triangle removal process.