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Hypergraph Independent Sets

Published online by Cambridge University Press:  11 October 2012

JONATHAN CUTLER  and
A. J. RADCLIFFE
JONATHAN CUTLER
Affiliation:
Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07043, USA (e-mail: jonathan.cutler@montclair.edu)
A. J. RADCLIFFE
Affiliation:
Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, USA (e-mail: aradcliffe1@math.unl.edu)
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Abstract

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal–Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.

Keywords

Primary 05C65Secondary 05D05, 05C69
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 1 , January 2013 , pp. 9 - 20
Copyright
Copyright © Cambridge University Press 2012

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