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Statement

Setting (pp. 2--3). For $\mathcal F\subset2^X$, $\mathcal F^\downarrow$ is the down-set generated by $\mathcal F$, the sets contained in some member of $\mathcal F$; it does not depend on $X$ (p. 2). Families $\mathcal F$ and $\mathcal G$ are cross-intersecting when $A\cap B\ne\emptyset$ for all $A\in\mathcal F$, $B\in\mathcal G$ (p. 3).

Theorem 6 (p. 3). If $\mathcal F,\mathcal G\subset2^{[n]}$ are cross-intersecting, then

$$|\mathcal G|+|\mathcal F|\le\max\{|\mathcal F^\downarrow|,|\mathcal G^\downarrow|\}. \tag{5}$$

The paper notes (p. 3) that when $\mathcal F$ and $\mathcal G$ are moreover both union families (no two members have union $[n]$), so are $\mathcal F^\downarrow$ and $\mathcal G^\downarrow$, the right side of (5) is then at most $2^{n-1}$, and $\mathcal F=\mathcal G$ gives the bound $2^{n-2}$ for IU families of Theorem 8.

Proof pointer

Section 3, p. 5. Assume $|\mathcal F^\downarrow|\le|\mathcal G^\downarrow|$. Theorem 5 matches $\mathcal F^\downarrow$ into $\mathcal G^\downarrow$ by disjoint pairs. Cross-intersection lets at most one set of each pair lie in its family ($\mathcal F$ or $\mathcal G$), which gives $|\mathcal F|+|\mathcal G|\le|\mathcal G^\downarrow|$.

Read depth

Claims checked: the definitions, Theorem 6 and the remark on union families were read clause by clause on the print, and the three-line proof on p. 5 was followed. Nothing here is independently reviewed.

Dependencies

Theorem 5.

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 6 on p. 3, its proof on p. 5. The edition is identified on the source card.

Bears on

• Problem 701: the paper's proof of Theorem 7, Chvátal's conjecture for intersecting families of covering number at most $2$, applies Theorem 6 to the two cross-intersecting traces of the family on a two-element cover. Theorem 6 by itself makes no statement about the conjecture.