Turán Problems and Shadows III: Expansions of Graphs

The expansion G of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a new vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let ex3(n, F ) denote the maximum number of edges in a 3-uniform hypergraph with n vertices not containing any copy of a 3-uniform hypergraph F . The study of ex3(n,G ) includes some well-researched problems, including the case that F consists of k disjoint edges [6], G is a triangle [5, 9, 18], G is a path or cycle [12, 13], and G is a tree [7, 8, 10, 11, 14]. In this paper we initiate a broader study of the behavior of ex3(n,G ). Specifically, we show ex3(n,K + s,t) = Θ(n 3−3/s) whenever t > (s− 1)! and s ≥ 3. One of the main open problems is to determine for which graphs G the quantity ex3(n,G ) is quadratic in n. We show that this occurs when G is any bipartite graph with Turán number o(n) where φ = 1+ √ 5 2 , and in particular, this shows ex3(n,G ) = O(n) when G is the three-dimensional cube graph.