Problem 701

Erdős Problem #701: declared status 'open'. Formalized: yes. Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever $\mathcal{F}'\subseteq \mathcal{F}$ is an intersecting subfamily we have$$\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.$$ Current best: A problem of Chv\'{a}tal [Ch74], who proved it replacing the closed under subsets condition with the (stronger) condition that, assuming all sets in $\mathcal{F}$ are subsets of $\{1,\ldots,n\}$, whenever $A\in \mathcal{F}$ and there is an injection $f:B\to A$ such that $x\leq f(x)$ for all $x\in B$, then $B\in \mathcal{F}$. Sterboul [St74] proved this when, letting $\mathcal{G}$ be the maximal sets (under inclusion) in $\mathcal{F}$, all sets in $\mathcal{G}$ have the same size, $\lvert A\cap B\rvert\leq 1$ for all $A\neq B\in \mathcal{G}$, and at least two sets in $\mathcal{G}$ have non-empty intersection. Prize: no. Tags: combinatorics, intersecting family.