On k-uniform tight cycles: the Ramsey number for C₊₍^ (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.