On the maximum number of distinct intersections in an intersecting family

08/01/2021

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by   Peter Frankl, et al.

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For n > 2k ≥ 4 we consider intersecting families ℱ consisting of k-subsets of {1, 2, …, n}. Let ℐ(ℱ) denote the family of all distinct intersections F ∩ F', F ≠ F' and F, F'∈ℱ. Let 𝒜 consist of the k-sets A satisfying |A ∩{1, 2, 3}| ≥ 2. We prove that for n ≥ 50 k^2 |ℐ(ℱ)| is maximized by 𝒜.