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 \(\lvert A\cap B\rvert=1\). A conjecture of Erdős and Sós. Katona (unpublished) proved this when \(k=4\), and Frankl [Fr77] proved this for all \(k\geq 4\). Frankl's theorem is for \(n\) sufficiently large in terms of \(k\), and this is the form stated here. Without this assumption the statement is false, see `erdos_702.variants.all_n_false`.