Gyárfás's conjecture on few-coloured matchings in uniform hypergraphs

Let $r$, $s$, $t$, $k$ be positive integers. A $t$-colouring assigns one of $t$ colours to every edge of a complete $r$-uniform hypergraph, and an $s$-coloured matching of size $k$ is a matching of $k$ edges whose edges use at most $s$ colours. Gyárfás's conjecture. For any $t$-colouring of the complete $r$-uniform hypergraph on

$$n \geq kr + \left\lfloor \frac{(k-1)(t-s)}{1+r+\dots+r^{s-1}} \right\rfloor$$

vertices, there exists an $s$-coloured matching of size $k$. The conjecture generalizes the known graph case to complete uniform hypergraphs; the paper proves the first nontrivial case for $3$-uniform hypergraphs and $3$ colours, supporting the conjecture.