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Uniformly cross intersecting families Eyal Lubetzky

Summary: Uniformly cross intersecting families
Noga Alon
Eyal Lubetzky
October 18, 2008
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 |AB| = 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 (2n
), by Frankl and
Ršodl. 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
= (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


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics