On independent sets in hypergraphs1. Issue 2 (2nd August 2012) - Kostochka, Alexandr
Abstract The independence numberα(H) 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 (r+1)‐uniform hypergraph in which every r‐element set is contained in at most d edges, where 0<d<n/(logn)3r2, then α(Hn)≥cr(ndlog⁡nd)1/r where cr>0 satisfies cr∼r/e as r→∞. 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 r≥2 and all values of d∈ℕ, there are infinitely many Hn such that α(Hn)≤br(ndlog⁡nd)1/r where br>0 depends only on r. In addition, for many values of d we show br∼cr as r→∞, 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