 
Summary: Uniformly cross intersecting families
Noga Alon
Eyal Lubetzky
October 18, 2008
Abstract
Let A and B denote two families of subsets of an nelement set. The pair (A, B) is said to be
crossintersecting iff AB = for all A A and B B. Denote by P(n) the maximum value
of AB over all such pairs. The best known upper bound on P(n) is (2n
), by Frankl and
Ršodl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n 2, a simple construction
of an crossintersecting pair (A, B) with AB = 2
2n2
= (2n
/
), 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
