Forbidding Just One Intersection

Abstract Following a conjecture of P. Erdos, we show that if F is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ⩾ 2l + 2 and n sufficiently large | F | ⩽ (k − l − 1n − l − 1) with equality holding if and only if F consists of all the k-sets containing a fixed (l + 1)-set. In general we show | F | ⩽ dknmax;{;l,k − l − 1};, where dk is a constant depending only on k. These results are special cases of more general theorems (Theorem 2.1–2.3).

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