Abstract: A basic problem in combinatorics is to determine the independence number of a hypergraph. We will consider this question in various contexts. Our results show that the behavior of the independence number of sparse hypergraphs is quite different than the graph case. We will also describe some hypergraph generalizations of the graph Ramsey number $R(3,t)$. Our results address old questions and conjectures of Erdos-Hajnal (1972) and of Ajtai-Erdos-Komlos-Szemeredi (1981).
