Independent sets in hypergraphs omitting an intersection

Abstract

AbstractA k‐uniform hypergraph with n vertices is an ‐omitting system if it has no two edges with intersection size . If in addition it has no two edges with intersection size greater than , then it is an ‐system. Rödl and Šiňajová proved a sharp lower bound for the independence number of ‐systems. We consider the same question for ‐omitting systems. Our proofs use adaptations of the random greedy independent set algorithm, and pseudorandom graphs. We also prove related results where we forbid more than two edges with a prescribed common intersection size leading to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number , where F is the k‐uniform Fan. The behavior is quite different than the case which is the classical Ramsey number .