On k-uniform Tight Cycles: the Ramsey Number for $$C_{kn}^{(k)}$$ and an Approximate Lehel’s Conjecture

Abstract 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.