Uniformly cross intersecting families

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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 AA and BB. 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 ≥ 2, a simple construction of an -cross-intersecting pair (A,B) with |A||B| = \( \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) \)2n−2 = Θ(2n/\( \sqrt \ell \)), and conjectured that this is best possible. Consequently, Sgall asked whether or not P e (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 e (n) ≤ \( \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) \)2n−2 for all 0. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.

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Correspondence to Noga Alon.

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Research supported in part by the Israel Science Foundation, by a USA-Israel BSF grant, by an ERC advanced grant, by NSF grant CCF 0832797 and by the Ambrose Monell Foundation.

Research partially supported by a Charles Clore Foundation Fellowship.

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Alon, N., Lubetzky, E. Uniformly cross intersecting families. Combinatorica 29, 389–431 (2009). https://doi.org/10.1007/s00493-009-2332-6

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Mathematics Subject Classification (2000)

  • 05D05