On k-uniform tight cycles: the Ramsey number for Cₖₙ⁽ᵏ⁾ and an approximate Lehel's conjecture

A k-uniform tight cycle is a k-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of k consecutive vertices in that ordering. We show that, for each k ≥ 3, the Ramsey number of the k-uniform tight cycle on $kn$ vertices is $(1+o(1))(k+1)n$. This is an extension to all uniformities of previous results for $k = 3$ by Haxell, {}uczak, Peng, R\"odl, Ruci\'nski, and Skokan and for $k = 4$ by Lo and the author and confirms a special case of a conjecture by the former set of authors. Lehel's conjecture, which was proved by Bessy and Thomass\'e, states that every red-blue edge-coloured complete graph contains a red cycle and a blue cycle that are vertex-disjoint and together cover all the vertices. We also prove an approximate version of this for k-uniform tight cycles. We show that, for every k ≥ 3, every red-blue edge-coloured complete k-graph on n vertices contains a red tight cycle and a blue tight cycle that are vertex-disjoint and together cover $n - o(n)$ vertices.