Let fm(a,b,c,d) denote the
maximum size family of a family
F of subsets of an m-element set so that there
is no pair A,B ∈ F with
|A∩ B|≥ a,
|Ac ∩ B|≥ b,
|A∩ Bc|≥ c,
|Ac∩ Bc|≥ 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
fm(0,b,b,0) is
Θ(mb) and
fm(a,0,0,d) is Θ(ma).
This can be viewed as a result
concerning forbidden configurations,
and provides further evidence for a conjecture of Anstee and Sali.
Our key tool is a strong stability version of the
Ahlswede-Khachatrian Complete Intersection Theorem, which
is of independent interest.
This is joint work with Richard Anstee.