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arXiv · 2501.07644

A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

Amarja Kathapurkar·Patrick Morris·Guillem Perarnau
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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.

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Amarja Kathapurkar, Patrick Morris, Guillem Perarnau. 2025-01-13. A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs. https://doi.org/10.1017/s0963548325100266

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