Partitioning 2-coloured complete k-uniform hypergraphs into monochromatic ℓ-cycles.

We show that for all ℓ,k,n with ℓ≤k∕2 and (k−ℓ) dividing n the following hypergraph-variant of Lehel’s conjecture is true. Every 2-edge-colouring of the k-uniform complete hypergraph Kn(k) on n vertices has at most two disjoint monochromatic ℓ-cycles in different colours that together cover all but at most 4(k−ℓ) vertices. If ℓ≤k∕3, then at most two ℓ-cycles cover all but at most 2(k−ℓ) vertices. Furthermore, we can cover all vertices with at most 4 (3 if ℓ≤k∕3) disjoint monochromatic ℓ-cycles.