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, G is a triangle, G is a path or cycle, and G is a tree. In this paper we initiate a broader study of the behavior of ex3(n,G⁺). Specifically, we show ex3(n,K⁺ s,t) = Φ(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(nΦ) where Φ = 1⁺ √5 2 , and in particular this shows ex3(n,G⁺) = O(n²) when G is the three-dimensional cube graph.