Pairwise intersections and forbidden configurations

Let f m ( 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 f m ( a , b , c , d ) is Θ ( m a + b − 1 ) if either b > c or a , b ≥ 1 . We also show that f m ( 0 , b , b , 0 ) is Θ ( m b ) and f m ( a , 0 , 0 , d ) is Θ ( m a ) . 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.