Uniformly cross intersecting families

Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be `-cross-intersecting iff |A∩B| = ` for all A ∈ A and B ∈ B. Denote by P`(n) the maximum value of |A||B| over all such pairs. The best known upper bound on P`(n) is Θ(2), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2`, a simple construction of an `-cross-intersecting pair (A,B) with |A||B| = ( 2` ` ) 2 = Θ(2/ √ `), and conjectured that this is best possible. Consequently, Sgall asked whether or not P`(n) decreases with `. In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large `, 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 `0 > 0, such that P`(n) ≤ ( 2` ` ) 2 for all ` ≥ `0. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.