On independent sets in hypergraphs

Abstract

AbstractThe 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 urn:x-wiley::media:rsa20453:rsa20453-math-0004 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 urn:x-wiley::media:rsa20453:rsa20453-math-0010 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