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Intersecting families are essentially contained in juntas. (English) Zbl 1243.05235
Summary: A family \(\mathcal J\) of subsets of \(\{1,\dots , n\}\) is called a \(j\)-junta if there exists \(J \subseteq \{1, \dots, n\}\), with \(|J| = j\), such that the membership of a set \(S\) in \(\mathcal J\) depends only on \(S \cap J\).
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 \(\mathcal A\) be a family of pairwise intersecting subsets of \(\{1, \dots, n\}\), all of size \(k\). We show that such a family is essentially contained in a \(j\)-junta \(\mathcal J\), 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 \(\mathcal A\) there exists an element \(i \in \{1, \dots, n\}\) such that the number of sets in \(\mathcal A\) that do not contain \(i\) is of order \(\binom{n-2}{k-2}\) (which is approximately \(\frac {k}{n-k}\) 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.

05D05 Extremal set theory
05A18 Partitions of sets
05B99 Designs and configurations
Full Text: DOI
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