Independence numbers of hypergraphs with sparse neighborhoods

Let H be a hypergraph with N vertices and average degree d. Suppose that the neighborhoods of H are sparse, then its independence number is at least cN(logd /d), where c>0 is a constant. In particular, let integers r≥3 and n≥1 be fixed, and let H be r-uniform, triangle-free and linear, then its independence number is at least cNlognd/d for all sufficiently large d.