A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

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.

💡 Research Summary

The paper addresses a central problem in extremal combinatorics: whether spanning structures that are guaranteed by Dirac‑type minimum‑degree conditions remain robust when the host hypergraph is coloured with a bounded number of edges per colour. This question is formalised in the “meta‑conjecture” of Coulson, Keevash, Perarnau and Yepremyan, which predicts that any spanning structure that appears above its extremal threshold can be found in a rainbow form in any suitably bounded colouring of the host (hyper)graph.

The authors focus on k‑uniform hypergraphs and the most natural spanning structure in this setting – a loose Hamilton cycle (ℓ=1). For a given integer j with 1≤j≤k−1, the j‑degree δ_j(G) of a hypergraph G is the minimum number of edges containing any fixed j‑set of vertices. The known extremal threshold δ_{k,j}(1) is the smallest real number such that every sufficiently large n‑vertex k‑graph with δ_j(G) ≥ (δ_{k,j}(1)+ε)n^{k−j} contains a loose Hamilton cycle. While the exact values of δ_{k,j}(1) are known only for j=k−1 (the codegree case) and j=k−2, the conjecture does not require knowledge of the exact constant.

Main Result (Theorem 1.2).
For any integers 1≤j<k and any ε>0 there exists a constant µ>0 (depending only on ε, k, j) such that the following holds for all sufficiently large n divisible by (k−1). Let G be an n‑vertex k‑uniform hypergraph with \