This research originated when the authors participated in the AIM workshop Hypergraph Turán Problem, March 21–25, 2011, Palo Alto, CA.
On independent sets in hypergraphs†
Article first published online: 2 AUG 2012
Copyright © 2012 Wiley Periodicals, Inc.
Random Structures & Algorithms
Volume 44, Issue 2, pages 224–239, March 2014
How to Cite
Kostochka, A., Mubayi, D. and Verstraëte, J. (2014), On independent sets in hypergraphs. Random Struct. Alg., 44: 224–239. doi: 10.1002/rsa.20453
Supported by NSF (DMS-0965587); Russian Foundation for Basic Research (09-01-00244-a).
Supported in part by NSF (DMS 0653946 and DMS 0969092).
Supported by NSF (DMS-0800704).
- Issue published online: 21 JAN 2014
- Article first published online: 2 AUG 2012
- Manuscript Accepted: 28 APR 2012
- Manuscript Revised: 7 DEC 2011
- Manuscript Received: 14 JUN 2011
- independent sets;
- steiner systems
The independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove that if Hn is an n-vertex -uniform hypergraph in which every r-element set is contained in at most d edges, where , then
where satisfies as . The value of cr improves and generalizes several earlier results that all use a theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi (J Comb Theory Ser A 32 (1982), 321–335). Our relatively short proof extends a method due to Shearer (Random Struct Algorithms 7 (1995), 269–271) and Alon (Random Struct Algorithms 9 (1996), 271–278). The above statement is close to best possible, in the sense that for each and all values of , there are infinitely many Hn such that
where depends only on r. In addition, for many values of d we show as , so the result is almost sharp for large r. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 224-239, 2014