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 \(k \ge 3\) , 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, Łuczak, Peng, Rödl, Ruciński, 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 Thomassé, 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 \(k \ge 3\) , 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.
