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Claim. Theorem 1 of the paper: there is $n_0$ such that for every $n\ge n_0$ and every $s$ with $1\le s\le(n-2)/3$, a $3$-uniform hypergraph on $n$ vertices whose largest matching has $s$ edges has at most $\max\{\binom n3-\binom{n-s}{3},\binom{3s+2}{3}\}$ edges, and every extremal hypergraph is the cover (all triples meeting a fixed $s$-set) or the clique (all triples inside a $(3s+2)$-set). In the notation of Problem 1020, with $k=s+1$,

$$f(n;3,k)=\max\left(\binom{3k-1}{3},\binom n3-\binom{n-k+1}{3}\right) \qquad(n\ge n_0,\ 3k-1\le n),$$

so the case $r=3$ holds for every $k$ once $n$ is large. The authors note that $n_0$ is not made effective and that the uniqueness fails at $n=6$, $s=1$. The paper is T. Łuczak and K. Mieczkowska, On Erdős' extremal problem on matchings in hypergraphs, J. Combin. Theory Ser. A 124 (2014), 178–194, carded at On Erdős' extremal problem on matchings in hypergraphs.

Covers. The case $r=3$ for $n\ge n_0$ and every $k$ with $3k-1\le n$, which the site records as $r=3$ for all $k$. The remaining $n<n_0$ are settled on [[problems/set_systems/E1020/claims/2012_05_30_frankl|Frankl 2017]], which proves the case $r=3$ for every $n\ge3k-1$.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Journal of Combinatorial Theory, Series A, 124 (2014), 178–194, after its first posting as arXiv:1202.4196 on 2012-02-19. The site labels the problem FALSIFIABLE, an open label, so its commentary, which credits the case to the paper as [LuMi14], is not acceptance and no reviewed is listed. Nothing here rests on this project's own review.