Wickets in 3-uniform Hypergraphs

Abstract

In these notes, we consider a Tur\'an-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let $H_n^{(3)}$ be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called {\em wicket}, is formed by three rows and two columns of a $3 \times 3$ point matrix. We describe two linear hypergraphs -- both containing a wicket -- that if we forbid either of them in $H_n^{(3)}$, then the hypergraph is sparse, and the number of its edges is $o(n^2)$. This proves a conjecture of Gy\'arf\'as and S\'ark\"ozy.