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Statement

Setting (pp. 1--3). A family is intersecting when any two of its members meet (p. 1). For a down-set $\mathcal D\subset2^X$, Chvátal's conjecture (Conjecture 1, p. 2) asserts that every intersecting $\mathcal F\subset\mathcal D$ satisfies

$$|\mathcal F|\le\max_{x\in X}\bigl|\{F\in\mathcal D:x\in F\}\bigr|=:\Delta(\mathcal D). \tag{3}$$

For a family $\mathcal F$ of non-empty sets, the covering number $\tau(\mathcal F)$ is the least $t$ such that some $t$-set meets every member of $\mathcal F$ (p. 3).

Theorem 7 (p. 3, quoted). "Suppose that $\mathcal F\subset\mathcal G\subset2^{[n]}$, $\mathcal G$ is a down-set and $\mathcal F$ is intersecting. If $\tau(\mathcal F)\le2$ then (3) holds."

Here (3) is read with $\mathcal D=\mathcal G$ and $X=[n]$: $|\mathcal F|\le\Delta(\mathcal G)$. The proof treats $\tau(\mathcal F)=2$; the case $\tau(\mathcal F)=1$, a family inside the star of one element, is immediate (an observation of this page).

Proof pointer

Section 3, pp. 5--6. Take the cover $\{1,2\}$. Split $\mathcal F$ into the traces on $2^{[3,n]}$ of its members meeting $[2]$ in $\{1\}$, in $\{2\}$ and containing both ($\mathcal F_1$, $\mathcal F_2$, $\mathcal F_{12}$), and split $\mathcal G$ likewise. Since $\mathcal F_{12}\subset\mathcal G_{12}$, it suffices to prove (8), $|\mathcal F_1|+|\mathcal F_2|\le\max\{|\mathcal G_1|,|\mathcal G_2|\}$. As $\mathcal F$ is intersecting, $\mathcal F_1$ and $\mathcal F_2$ are cross-intersecting, and Theorem 6 bounds their total by $\max\{|\mathcal F_1^\downarrow|,|\mathcal F_2^\downarrow|\}$, which is at most $\max\{|\mathcal G_1|,|\mathcal G_2|\}$. The print justifies the last step by calling $\mathcal F_1,\mathcal F_2$ down-sets; what the step uses is $\mathcal F_i^\downarrow\subset\mathcal G_i$, which holds because $\mathcal G$ is a down-set (an observation of this page).

Read depth

Claims checked: Conjecture 1, the definition of $\tau$, Theorem 7 and its proof on pp. 5--6 were read clause by clause on the print. Nothing here is independently reviewed.

Dependencies

Theorem 6, and through it 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, Conjecture 1 on p. 2, Theorem 7 on p. 3, its proof on pp. 5--6. The edition is identified on the source card.

Bears on

• Problem 701: Conjecture 1, which the paper credits to Chvátal and states for a down-set in $2^X$, is the problem's corrected Statement when $X$ is finite. Theorem 7 proves its inequality for the intersecting subfamilies of covering number at most $2$; a single element of largest degree in $\mathcal G$ serves for all of them. The covering condition is on the intersecting subfamily, not on the down-set. For larger covering number the paper has only the weaker bound $|\mathcal F|\le|\mathcal G|/2$ of Theorem 3.