The independence number of a hypergraph H is the maximum size of an independent set of vertices of H – a set of vertices containing no edges of H. The problem of determining the maximum size of an independent set in an r-uniform hypergraph with a prescribed number of vertices and edges is equivalent to the notorious Turán problem for hypergraphs. In this chapter, we consider the fundamental problem of lower bounds on the independence number of H when H is an n-vertex r-uniform linear hypergraph of average degree d, with an illustration of applications to the Erdős–Heilbronn triangle problem and to lower bounds on extremal numbers for bipartite graphs. In particular, we give a novel proof of the Ajtai–Komlós–Pintz–Spencer– Szemerédi Theorem, giving large independent sets in uncrowded hypergraphs.