---

Claim. Theorem 1 of the paper, in the problem's notation: if $r\ge2$, $k\ge2$, $n>2r^3(k-1)$ and an $r$-uniform hypergraph on $n$ vertices has no $k$ pairwise disjoint edges and more than $\binom nr-\binom{n-k+1}{r}-\binom{n-k-r+1}{r-1}+1$ edges, then some $(k-1)$-set of vertices meets every edge, so the hypergraph is contained in the covering family $E_r(n,k-1)$ of all $r$-sets meeting a fixed $(k-1)$-set. In particular

$$f(n;r,k)=\binom nr-\binom{n-k+1}{r}\qquad(n>2r^3(k-1)),$$

the conjectured value of Problem 1020 in that range, with the covering family the unique extremal hypergraph and a stability statement beside it. The paper writes $k$ for the matching number allowed, the problem's $k-1$. The theorem extends the Hilton–Milner theorem, its case of one allowed edge, to every matching number, and makes Erdős's 1965 range explicit; the proof is an induction on the matching number through a degree lemma. The paper is B. Bollobás, D. E. Daykin and P. Erdős, Sets of independent edges of a hypergraph, Quart. J. Math. Oxford Ser. (2) 27 (1976), 25–32, carded at Sets of independent edges of a hypergraph.

Covers. The range $n>2r^3(k-1)$, which the site records as $n\ge2kr^3$. The range was widened to order $r^2k$ on Huang, Loh and Sudakov 2012 and Frankl, Łuczak and Mieczkowska 2012, and to order $rk$ on Frankl 2013.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Quarterly Journal of Mathematics, Oxford Second Series, in 1976 (volume 27, issue 1); the record gives only the year, so the page is dated to its first day. The site labels the problem FALSIFIABLE, an open label, so its commentary, which credits the range to the paper as [BDE76], is not acceptance and no reviewed is listed. Nothing here rests on this project's own review.