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Intersecting Families are Essentially Contained in Juntas

Published online by Cambridge University Press:  01 March 2009

IRIT DINUR  and
EHUD FRIEDGUT
IRIT DINUR
Affiliation:
School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel (e-mail: dinuri@cs.huji.ac.il)
EHUD FRIEDGUT
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, Israel (e-mail: ehudf@math.huji.ac.il)
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Abstract

A family of subsets of {1, . . ., n} is called a j-junta if there exists J ⊆ {1, . . ., n}, with |J| = j, such that the membership of a set S in depends only on S ∩ J.

In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let be a family of pairwise intersecting subsets of {1, . . ., n}, all of size k. We show that such a family is essentially contained in a j-junta , where j does not depend on n but only on the ratio k/n and on the interpretation of ‘essentially’.

When k = o(n) we prove that every intersecting family of k-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family there exists an element i ∈ {1, . . ., n} such that the number of sets in that do not contain i is of order (which is approximately times the size of a maximal intersecting family).

Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 1-2: Papers from The 2006 Oberwolfach Meeting on Combinatorics, Probability and Computing , March 2009 , pp. 107 - 122
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Ahlswede, R. and Khachatrian, L. (1997) The complete intersection theorem for systems of finite sets. Europ. J. Combin. 18 125–136.CrossRefGoogle Scholar
[2]Dinur, I., Friedgut, E. and Regev, O. Independent sets in graph powers are almost contained in juntas. To appear in G.A.F.A.Google Scholar
[3]Dinur, I. and Safra, S. (2005) On the hardness of approximating minimum vertex cover. Ann. of Math. 162 439–485.CrossRefGoogle Scholar
[4]Erdős, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2 12 313–318.CrossRefGoogle Scholar
[5]Frankl, P. and Tokushige, N. (1998) Some inequalities concerning cross-intersecting families. Combin. Probab. Comput. 7 247–260.CrossRefGoogle Scholar
[6]Friedgut, E. (1998) Boolean functions with low average sensitivity depend on few coordinates. Combinatorica 18 27–35.CrossRefGoogle Scholar
[7]Füredi, Z. (1995) Cross-intersecting families of finite sets. J. Combin. Theory Ser. A 72 332–339.CrossRefGoogle Scholar
[8]Katona, G. O. H. (1968) A theorem of finite sets. In Theory of Graphs (Erdőos, P. and Katona, G., eds), Akadémiai Kiadó and Academic Press, pp. 187–207.Google Scholar
[9]Kruskal, J. (1963) The number of simplices in a complex. In Mathematical Optimization Techniques (Bellman, R., ed.), University of California Press, pp. 251–278.CrossRefGoogle Scholar
[10]Margulis, G. (1974) Probabilistic characteristics of graphs with large connectivity. Prob. Peredachi Inform. 10 101–108.Google Scholar
[11]Mossel, E., O'Donnell, R. and Oleszkiewicz, K. (2005) Noise stability of functions with low influences: Invariance and optimality. In Proc. 46th IEEE Symp. on Foundations of Computer Science, pp. 21–30.CrossRefGoogle Scholar
[12]Russo, L. (1982) An approximate zero-one law. Z. Wahrsch. werw. Gebiete 61 129–139.CrossRefGoogle Scholar
[13]Tuza, Z. (1985) Critical hypergraphs and intersecting set-pair systems. J. Combin. Theory Ser. B 39 134–145.CrossRefGoogle Scholar