Uniformly cross intersecting families

Abstrac: Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff {pipe}A∩B{pipe} = ℓ for all A ∈ A and B ∈ B. Denote by P_ℓ(n) the maximum value of {pipe}A{pipe}{pipe}B{pipe} over all such pairs. The best known upper bound on P_ℓ(n) is Θ(2ⁿ), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with, and conjectured that this is best possible. Consequently, Sgall asked whether or not P_ℓ(n) decreases with ℓ. In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large ℓ, implying a positive answer to the above question of Sgall as well. By analyzing the linear spaces of the characteristic vectors of A, B over ℝ, we show that there exists some ℓ₀ > 0, such that, for all ℓ ≥ ℓ₀. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.