Structure and stability of triangle-free set systems

Author:
Dhruv Mubayi

Journal:
Trans. Amer. Math. Soc. **359** (2007), 275-291

MSC (2000):
Primary 05C35, 05C65, 05D05

DOI:
https://doi.org/10.1090/S0002-9947-06-04009-8

Published electronically:
August 16, 2006

MathSciNet review:
2247891

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Abstract | References | Similar Articles | Additional Information

Abstract: We define the notion of stability for a monotone property of set systems. This phenomenon encompasses some classical results in combinatorics, foremost among them the Erdos-Simonovits stability theorem. A triangle is a family of three sets such that , , are each nonempty, and . We prove the following new theorem about the stability of triangle-free set systems.

Fix . For every , there exist and such that the following holds for all : if and is a triangle-free family of -sets of containing at least members, then there exists an -set which contains fewer than members of .

This is one of the first stability theorems for a nontrivial problem in extremal set theory. Indeed, the corresponding extremal result, that for every triangle-free family of -sets of has size at most , was a longstanding conjecture of Erdos (open since 1971) that was only recently settled by Mubayi and Verstraëte (2005) for all .

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Additional Information

**Dhruv Mubayi**

Affiliation:
Department of Mathematics, Statistics, & Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045

Email:
mubayi@math.uic.edu

DOI:
https://doi.org/10.1090/S0002-9947-06-04009-8

Keywords:
Extremal set theory,
intersecting family,
stability theorems

Received by editor(s):
March 16, 2004

Received by editor(s) in revised form:
November 3, 2004

Published electronically:
August 16, 2006

Additional Notes:
The author was supported in part by NSF Grant DMS #0400812 and by an Alfred P. Sloan Research Fellowship.

Article copyright:
© Copyright 2006
by the author