Towards Lehel's conjecture for 4-uniform tight cycles

A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size k formed by k consecutive vertices in the ordering. We prove that every red-blue edge-coloured K_n⁽⁴⁾ contains a red and a blue tight cycle that are vertex-disjoint and together cover n−o(n) vertices. Moreover, we prove that every red-blue edge-coloured K_n⁽⁵⁾ contains four monochromatic tight cycles that are vertex-disjoint and together cover n−o(n) vertices.