A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

Abstract

A meta-conjecture of Coulson, Keevash, Perarnau, and Yepremyan [12] 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≤ j ≤ 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.