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 |A∩B|=ℓ for all A∈ A and B∈B. Denote by P _e (n) the maximum value of |A||B| over all such pairs. The best known upper bound on P _e (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 |A||B| = $\left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right)$2^n−2ℓ = Θ(2ⁿ/$\sqrt \ell$), and conjectured that this is best possible. Consequently, Sgall asked whether or not P _e (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 ℓ ₀ > 0, such that P _e (n) ≤ $\left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right)$2^n−2ℓ for all ℓ≥ℓ ₀. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.