Erdős matching conjecture for uniformity at least four

Prove the Erdős Matching Conjecture for every uniformity $r\ge 4$, namely, establish that for the complete $r$-uniform hypergraph $K_{1,\ldots,1}^{r}$ and the stated range of $n$ and $k$, the extremal number of an $(k+1)$-matching equals $\max\{\binom{n}{r}-\binom{n-k}{r},\binom{r(k+1)-1}{r}\}$.

Background

The Erdős Matching Conjecture concerns the maximum number of edges in an $n$-vertex $r$-uniform hypergraph whose matching number is at most $k$. It predicts that the extremal hypergraph is given by the larger of two standard constructions: all $r$-sets meeting a fixed $k$-vertex set, or the complete $r$-graph on $r(k+1)-1$ vertices.

The paper notes that the conjecture has been resolved for $r=3$, both asymptotically for large $n$ and in a later result for all $n$. The general case of uniformity at least four remains unresolved, making it a central open problem related to the tiling framework studied in the paper.