Tight Hamilton cycles with high discrepancy

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k\ge 3$ and $r\ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$ with minimum $(k-1)$-degree $\delta (G)\ge (1\mathrm{/}2+o(1))n$ contains a tight Hamilton cycle with high discrepancy, that is, with at least $n\mathrm{/}r+\mathrm{\Omega }(n)$ edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Turán-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.