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(4)n 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(5)n contains four monochromatic tight cycles that are vertex-disjoint and together cover n−o(n) vertices.