Wickets in 3-uniform hypergraphs

Abstract

In this note, we consider a Turán-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let Hn(3) be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called wicket, is formed by three rows and two columns of a 3×3 point matrix. We describe two linear hypergraphs that if we forbid either of them in Hn(3), then the hypergraph is sparse, i.e. the number of its edges is o(n2). Since both contain a wicket, it implies a conjecture of Gyárfás and Sárközy that wicket-free hypergraphs are sparse.