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A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs
Published online by Cambridge University Press: 11 November 2025
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 
$1\leq j\leq k-1$, 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.
MSC classification
Secondary:
05C65: Hypergraphs
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- © The Author(s), 2025. Published by Cambridge University Press
Footnotes
*
Research supported by EPSRC Research grant EP/R034389/1.
†
Research supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Walter Benjamin programme - project number 504502205 and by the European Union’s Horizon Europe Marie Sk lodowska-Curie grant RAND-COMB-DESIGN - project number 101106032 
.
‡
Research supported by the grants RED2022-134947-T, PID2023-147202NB-I00, PCI2024-155080-2 and the Programme Severo Ochoa y María de Maeztu por Centros y Unidades de Excelencia en I&D (CEX2020-001084-M), all of them funded by MICIU/AEI/10.13039/501100011033.
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