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Statement

Setting. Cross-IU is defined on the Theorem 9 page: every member of one family meets every member of the other, and no such pair has union $[n]$.

Theorem 10 (p. 3). If $\mathcal A_1,\mathcal A_2,\dots,\mathcal A_d\subset2^{[n]}$ are pairwise cross-IU, then

$$\sum_{i=1}^d|\mathcal A_i|\le\max\{2^n,\,d\cdot2^{n-2}\}.$$

The paper adds (p. 3, quoted) that "the equality holds only if $d\le4$ and $\mathcal A_1=2^{[n]}$ [sic] for some $i$ or $d\ge4$ and $\mathcal A_1=\ldots=\mathcal A_d$", the index $1$ standing where $i$ is evidently meant. It compares the result with Hilton's theorem for pairwise intersecting families of $k$-sets.

Theorem 12 (p. 7). If $\mathcal A,\mathcal B\subset2^{[n]}$ are cross-IU and $|\mathcal A|\ge|\mathcal B|$, then

$$|\mathcal A|+3|\mathcal B|\le2^n. \tag{14}$$

Corollary 1 (p. 6). If $\mathcal A_1,\dots,\mathcal A_d\subset2^{[n]}$ are pairwise cross-IU, $d\ge5$ and $|\mathcal A_1|\ge|\mathcal A_2|\ge\cdots\ge|\mathcal A_d|$, then

$$|\mathcal A_1|+\cdots+|\mathcal A_d|\le d\,2^{n-2}, \tag{13}$$

with strict inequality unless $\mathcal A_1=\cdots=\mathcal A_d$.

Lemma 1 (p. 6). If $d\ge1$ and $1\le x\le d$, then $x+d/x\le1+d$ (12).

The paper says Theorem 10 follows at once from Corollary 1 and Theorem 12. For $d\ge5$ this is Corollary 1, after ordering the families by size. For $2\le d\le4$, ordering the families by size, Theorem 12 applied to the two largest gives $\sum_i|\mathcal A_i|\le|\mathcal A_1|+3|\mathcal A_2|\le2^n$ (the step spelled out on this page; $d=1$ is trivial).

Proof pointer

Corollary 1, pp. 6--7: with $a_i=|\mathcal A_i|/2^{n-2}$, (7) of Theorem 9 gives $a_i\le1/a_1$ for $i\ge2$, and Lemma 1 gives $a_1+(d-1)/a_1\le d$ when $1\le a_1\le4$; adding single sets to the families handles the equality case. Theorem 12, p. 7: with $x=|\mathcal A|/2^n$, $y=|\mathcal B|/2^n$, (7) gives $xy\le1/16$, which settles $x\le3/4$; for $x=1-z$ with $0\le z\le1/4$, Harris–Kleitman applied to the generated up-sets and down-sets gives $y\le z^2(1-z)<z/4$.

Read depth

Claims checked: Theorem 10 with its equality remark, Lemma 1, Corollary 1 and Theorem 12 were read clause by clause on the print, and their proofs on pp. 6--7 were followed for their structure and not checked line by line. Nothing here is independently reviewed.

Dependencies

Theorem 9; external input named by the paper: the Harris–Kleitman inequality (its Theorem 2, p. 2).

Source. P. Frankl and A. Kupavskii, Perfect matchings in down-sets, Discrete Math. 346 (2023), Paper No. 113323, DOI 10.1016/j.disc.2023.113323; read in arXiv:2201.03865v1, Theorem 10 on p. 3, Lemma 1 and Corollary 1 on p. 6, Theorem 12 on p. 7, with proofs on pp. 6--7. The edition is identified on the source card.

Bears on

No Erdős problem in the corpus is linked to these bounds.