Erdős Problem #702

Let $k\geq 4$. If $\mathcal{F}$ is a family of subsets of $\{1,\ldots,n\}$ with $\lvert A\rvert=k$ for all $A\in \mathcal{F}$ and $\lvert \mathcal{F}\rvert >\binom{n-2}{k-2}$ then there are $A,B\in\mathcal{F}$ such that $\lvert A\cap B\rvert=1$.

#702: [Er75f,p.108][Er81][Er82e]

combinatorics

A conjecture of Erdős and Sós. Katona (unpublished) proved this when $k=4$, and Frankl [Fr77] proved this for all $k\geq 4$.

See also [703].

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This page was last edited 16 October 2025.

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