Almost intersecting families

04/18/2020

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by   Peter Frankl, et al.

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Let n > k > 1 be integers, [n] = {1, …, n}. Let ℱ be a family of k-subsets of [n]. The family ℱ is called intersecting if F ∩ F' ≠∅ for all F, F' ∈ℱ. It is called almost intersecting if it is not intersecting but to every F ∈ℱ there is at most one F'∈ℱ satisfying F ∩ F' = ∅. Gerbner et al. proved that if n ≥ 2k + 2 then |ℱ| ≤n - 1 k - 1 holds for almost intersecting families. The main result implies the considerably stronger and best possible bound |ℱ| ≤n - 1 k - 1 - n - k - 1 k - 1 + 2 for n > (2 + o(1))k.