PROVED
This has been solved in the affirmative.
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 $\lvert A\cap B\rvert=1$.
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 Katona (unpublished) proved this when $k=4$, and Frankl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.
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #702, https://www.erdosproblems.com/702, accessed 2025-11-16
