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Claim. Kleitman's paper bounds the number of members of a family of subsets of an $n$-element set that contains no $k$ pairwise disjoint members, and shows the bounds best possible when $n=mk$ and when $n=mk-1$. Its $r$-uniform case at $n=rk$ is the case of Problem 1020 that the site and the later literature credit to the paper: a family of $r$-subsets of an $rk$-set with no $k$ pairwise disjoint members has at most

$$\binom{rk-1}{r}=\Bigl(1-\frac1k\Bigr)\binom{rk}{r}$$

members, so $f(rk;r,k)=\binom{rk-1}{r}$, the clique term of the conjecture, which at $n=rk$ equals the covering term. Kolupaev and Kupavskii record that at $n=(s+1)k$ in their notation the conjectured inequality is easy and was proved by Kleitman, which is why Frankl's range on Frankl 2017 needs no lower bound on the matching number. The paper is D. J. Kleitman, Maximal number of subsets of a finite set no $k$ of which are pairwise disjoint, J. Combinatorial Theory 5 (1968), 157–163.

Covers. The case $n=rk$ for every $r\ge3$ and $k\ge2$. It says nothing about other $n$; the neighboring range above $rk$ is the claim on Frankl 2017.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Journal of Combinatorial Theory in 1968 (volume 5, issue 2); the record dates the issue to September 1968 and gives no day, so the page is dated to the first day of that month. The site labels the problem FALSIFIABLE, an open label, so its commentary, which credits the case $n=kr$ to the paper as [Kl68], is not acceptance and no reviewed is listed. Nothing here rests on this project's own review.