Erdős Problem #21

The problem

Let $f(n)$ be minimal such that there is an intersecting family $\mathcal{F}$ of sets of size $n$ (so $A\cap B\neq\emptyset$ for all $A,B\in \mathcal{F}$) with $\lvert \mathcal{F}\rvert=f(n)$ such that any set $S$ with $\lvert S\rvert \leq n-1$ is disjoint from at least one $A\in \mathcal{F}$.

Is it true that

$$f(n) \ll n?$$

Statement quoted from erdosproblems.com/21, maintained by Thomas Bloom. Original source: [Er81][Er90][Er92b][Er97f]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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