## 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*(2^{n}), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all *n* ≥ 2*ℓ*, 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*(2^{n}/\( \sqrt \ell \)), 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.

This is a preview of subscription content, access via your institution.

## References

- [1]
R. Ahlswede, N. Cai and Z. Zhang: A general 4-words inequality with consequences for 2-way communication complexity,

*Adv. in Appl. Math.***10**(1989), 75–94. - [2]
N. Alon and J. H. Spencer:

*The Probabilistic Method*, Second Edition, Wiley, New York, 2000. - [3]
L. Babai and P. Frankl:

*Linear Algebra Methods in Combinatorics*, Preliminary Version 2, Dept. of Computer Science, The Univesity of Chicago, 1992. - [4]
P. Erdős: On a lemma of Littlewood and Offord,

*Bull. Amer. Math. Soc. (2nd ser.)***51**(1945), 898–902. - [5]
P. Erdős: Problems and results in graph theory and combinatorial analysis, in:

*Proc. of the Fifth British Comb. Conf. 1975 Aberdeen*, pp. 169–192, Congressus Numerantium, No.**XV**, Utilitas Math., Winnipeg, Man., 1976. - [6]
P. Erdős, C. Ko and R. Rado: Intersection theorems for systems of finite sets,

*Quart. J. Math. Oxford Ser. 2***12**(1961), 313–320. - [7]
P. Frankl: Extremal set systems, in:

*R. L. Graham, M. Grötschel, L. Lovász (eds.)*,*Handbook of Combinatorics*, Vol.**1, 2**, pp. 1293–1329, Elsevier, Amsterdam, 1995. - [8]
P. Frankl and Z. Füredi: Forbidding just one intersection,

*J. Combin. Theory Ser. A***39(2)**(1985), 160–176. - [9]
P. Frankl and V. Rödl: Forbidden intersections,

*Trans. Amer. Math. Soc*.**300**(1987), 259–286. - [10]
P. Frankl and R. M. Wilson: Intersection theorems with geometric consequences,

*Combinatorica***1(4)**(1981), 357–368. - [11]
G. Katona: Intersection theorems for systems of finite sets,

*Acta Math. Acad. Sci. Hungar*.**15**(1964), 329–337. - [12]
P. Keevash and B. Sudakov: On a restricted cross-intersection problem,

*J. Combinatorial Theory Ser. A***113**(2006), 1536–1542. - [13]
J. Littlewood and C. Offord: On the number of real roots of a random algebraic equation III,

*Mat. Sbornik***12**(1943), 277–285. - [14]
D. Lubell: A short proof of Sperner’s theorem,

*Journal of Combinatorial Theory***1**(1966), 299. - [15]
D. K. Ray-Chaudhuri and R. M. Wilson: On

*t*-designs,*Osaka J. Math*.**12**(1975), 737–744. - [16]
H. Robbins: A remark on Stirling’s formula,

*Amer. Math. Monthly***62**(1955), 26–29. - [17]
J. Sgall: Bounds on pairs of families with restricted intersections,

*Combinatorica***19(4)**(1999), 555–566. - [18]
E. Sperner: Ein Satz über Untermengen einer endlichen Menge,

*Math. Z*.**27**(1928), 544–548.

## Author information

### Affiliations

### Corresponding author

## Additional information

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.

## Rights and permissions

## About this article

### Cite this article

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

Received:

Published:

Issue Date:

### Mathematics Subject Classification (2000)

- 05D05