## Intersecting families of permutations

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- by David Ellis, Ehud Friedgut and Haran Pilpel
- J. Amer. Math. Soc.
**24**(2011), 649-682 - DOI: https://doi.org/10.1090/S0894-0347-2011-00690-5
- Published electronically: January 31, 2011
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## Abstract:

A set of permutations $I \subset S_n$ is said to be $k$-*intersecting*if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb {N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.

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## Bibliographic Information

**David Ellis**- Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB England
- Address at time of publication: St John’s College, Cambridge, CB2 1TP, United Kingdom
**Ehud Friedgut**- Affiliation: Department of Mathematics, Hebrew University, 91904 Jerusalem, Israel, and Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada
**Haran Pilpel**- Affiliation: Department of Mathematics, Hebrew University, 91904 Jerusalem, Israel
- Address at time of publication: Google, Inc., Levinstein Tower 26th Floor, 23 Manachem Begin St, 66183 Tel Aviv, Israel
- Received by editor(s): March 9, 2009
- Received by editor(s) in revised form: November 15, 2010, and December 8, 2010
- Published electronically: January 31, 2011
- Additional Notes: Research of the second author was supported in part by the Israel Science Foundation, grant no. 0397684, and NSERC grant 341527.

Research of the third author was supported in part by the Giora Yoel Yashinsky Memorial Grant. - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**24**(2011), 649-682 - MSC (2010): Primary 05E10, 20C30, 05D99
- DOI: https://doi.org/10.1090/S0894-0347-2011-00690-5
- MathSciNet review: 2784326