Erdős Problem 1020

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.