Bounding the Independence Number in Some (n,k,ℓ,λ)-Hypergraphs

Given positive integers k , ℓ and λ such that 2 ⩽ ℓ ⩽ k - 1 , an ( n , k , ℓ , λ ) -hypergraph H is a k -uniform hypergraph on the vertex set [ n ] in which every subset of size ℓ is contained in at most λ edges. The independence number α ( H ) of H is the maximum size of a subset of vertices which contains no edges of H . Let f ( n , k , ℓ , λ ) = min { α ( H ) ∣ H is an ( n , k , ℓ , λ ) -hypergraph } . In this paper we show that for any given positive integers k ⩾ 5 , 2 k + 4 5 < ℓ ⩽ k - 2 and λ ⩽ n 5 ℓ - 2 k - 4 3 k - 9 ω ( n ) , f ( n , k , ℓ , λ ) ⩾ C k , ℓ n λ log n λ 1 ℓ , where ω ( n ) → ∞ arbitrarily slowly as n → ∞ and C k , ℓ is a constant depending only on k and ℓ . In particular, C k , ℓ ∼ ℓ - 1 e as k → ∞ . It generalizes the results from the case ℓ = k - 1 or λ = 1 . An upper bound on f ( n , k , ℓ , λ ) is also obtained for k ⩾ 3 , 2 ⩽ ℓ ⩽ k - 1 and log n ≪ λ ≪ n , by considering a random k -uniform hypergraph H k ( n , p ) .