Pairwise intersections and forbidden configurations

Let fm(a, b, c, d) denote the maximum size of a family F of subsets of an m-element set for which there is no pair of subsets A, B ∈ F with |A ∩ B| ≥ a, |A- ∩ B| ≥ b, |A ∩ B-| ≥ c, and |A- ∩ B-| ≥ d. By symmetry we can assume a ≥ d and b ≥ c. We show that fm(a, b, c, d) is Θ(ma+b-1) if either b > c or a, b ≥ 1. We also show that fm(0, b, b, 0) is Θ(mb) and fm(a, 0, 0, d) is Θ(ma). The asymptotic results are as m → ∞ for fixed non-negative integers a, b, c, d. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.

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