PROVED
This has been solved in the affirmative.
Suppose $n\geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertices are $t$-coloured. Prove that some colour class must contain $k$ pairwise disjoint edges.
In other words, this problem asks to determine the chromatic number of the Kneser graph. This would be best possible: if $n=kr-1+(t-1)(k-1)$ then decomposing $[n]$ as one set $X_1$ of size $kr-1$ and $t-1$ sets $X_2,\ldots,X_{t}$ of size $k-1$, a colouring without $k$ pairwise disjoint edges is given colouring all subsets of $X_0$ in colour $1$ and assigning an edge with colour $2\leq i\leq t$ if $i$ is minimal such that $X_i$ intersects the edge.
When $k=2$ this was conjectured by Kneser and proved by Lovász
[Lo78]. The general case was proved by Alon, Frankl, and Lovász
[AFL86].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #780, https://www.erdosproblems.com/780, accessed 2025-12-07
