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Claim. The maximum number of edges of a $3$-uniform hypergraph on $n$ vertices with matching number $s$ is $\max\{\binom{3s+2}{3},\binom n3-\binom{n-s}{3}\}$ for all $n$ and $s$ with $n\ge3s+2$, as the paper's abstract states its main theorem. In the notation of Problem 1020, with $k=s+1$,

$$f(n;3,k)=\max\left(\binom{3k-1}{3},\binom n3-\binom{n-k+1}{3}\right) \qquad(n\ge3k-1),$$

which is the whole range in which the conjecture is meaningful at $r=3$, so the case $r=3$ of the conjecture is settled in full. The paper is P. Frankl, On the maximum number of edges in a hypergraph with given matching number, Discrete Appl. Math. 216 (2017), 562–581.

Covers. The case $r=3$ for every $n$ and $k$. Its predecessors are Frankl, Rödl and Ruciński 2012 for $n\ge4k$ and Łuczak and Mieczkowska 2014 for $n$ large. It says nothing about $r\ge4$.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in Discrete Applied Mathematics 216 (2017), 562–581, after its first posting as arXiv:1205.6847 on 2012-05-30. The site's commentary does not cite the paper and labels the problem FALSIFIABLE, an open label, so no reviewed is listed. Nothing here rests on this project's own review.