A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

TL;DR

This paper proves a rainbow version of a Dirac-type theorem for loose Hamilton cycles in hypergraphs, confirming a key case of a broader conjecture about spanning structures in colored hypergraphs.

Contribution

It establishes that above the degree threshold for loose Hamilton cycles, any bounded coloring contains a rainbow loose Hamilton cycle, solving a significant case of the meta-conjecture.

Findings

Proves a rainbow Dirac theorem for loose Hamilton cycles in hypergraphs.

Confirms a key case of the Coulson-Keevash-Perarnau-Yepremyan conjecture.

Shows that bounded colorings above the degree threshold contain rainbow cycles.

Abstract

A meta-conjecture of Coulson, Keevash, Perarnau and Yepremyan states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture, by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.