On independent sets in hypergraphs

The independence number α(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 H_n 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 c_r 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 H_n 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.