Erdős Problem #1020
Erdős matching conjecture

Let \(f(n;r,k)\) be the maximal number of edges in an \(r\)-uniform hypergraph which contains no set of \(k\) many independent edges. For all \(r\geq 3\), \[f(n;r,k)=\max\left(\binom{rk-1}{r}, \binom{n}{r}-\binom{n-k+1}{r}\right).\] Note: the source states the formula with no range on `n` or `k`, but some restriction is needed: e.g. for `r = 3`, `k = 2`, `n = 4` no two disjoint triples fit in `4` vertices, so the left-hand side is `4.choose 3 = 4` while the right-hand side is `5.choose 3 = 10`. We require `k ≥ 1` and `n ≥ r*k - 1`: this is the smallest `n` accommodating the construction counted by the first term (all `r`-subsets of a fixed `(r*k - 1)`-set), and at `n = r*k - 1` the equality holds trivially, since the complete `r`-uniform hypergraph has no `k`-matching. The source's commentary likewise calls the case `n < k*r` trivial.