Partitioning $2$-coloured complete $k$-uniform hypergraphs into monochromatic $\ell$-cycles

TL;DR

This paper proves a hypergraph version of Lehel's conjecture, showing that in any 2-colouring of a complete k-uniform hypergraph, most vertices can be covered by a small number of monochromatic -cycles, depending on parameters.

Contribution

It establishes new bounds on covering vertices with monochromatic -cycles in 2-coloured complete hypergraphs, extending classical conjectures to hypergraph settings.

Findings

At most two monochromatic -cycles cover all but a bounded number of vertices.

The bounds depend on the relation between  and k, with tighter bounds when   k/3.

All vertices can be covered with at most 4 monochromatic -cycles, or 3 if   k/3.

Abstract

We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$ vertices has at most two disjoint monochromatic $\ell$-cycles in different colours that together cover all but at most $4(k-\ell)$ vertices. If $\ell \leq k/3$, then at most two $\ell$-cycles cover all but at most $2(k-\ell)$ vertices. Furthermore, we can cover all vertices with at most $4$ ($3$ if $\ell\leq k/3$) disjoint monochromatic $\ell$-cycles.