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Turán problems and shadows III: expansions of graphs
, 2015
"... The expansion G+ of a graph G is the 3uniform 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 3uniform hypergraph with n vertices not conta ..."
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The expansion G+ of a graph G is the 3uniform 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 3uniform hypergraph with n vertices not containing any copy of a 3uniform hypergraph F. The study of ex3(n,G +) includes some wellresearched 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
A survey of Turán problems for expansions
, 2015
"... The rexpansion G+ of a graph G is the runiform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n ..."
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The rexpansion G+ of a graph G is the runiform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n vertices not containing any copy of the runiform hypergraph F. Many problems in extremal set theory ask for the determination of exr(n,G +) for various graphs G. We survey these Turántype problems, focusing on recent developments. 1
Turán problems and shadows II: trees
, 2014
"... The expansion G+ of a graph G is the 3uniform hypergraph obtained from G by enlarging each edge of G with a vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n vertices not contai ..."
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The expansion G+ of a graph G is the 3uniform hypergraph obtained from G by enlarging each edge of G with a vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n vertices not containing any copy of F. The authors [11] recently determined ex3(n,G +) more generally, namely when G is a path or cycle, thus settling conjectures of FürediJiang [9] (for cycles) and FürediJiangSeiver [10] (for paths). Here we continue this project by determining the asymptotics for ex3(n,G +) when G is any fixed forest. This settles a conjecture of Füredi [8]. Using our methods, we also show that for any graph G, either ex3(n,G +) ≤ ( 12 + o(1))n2 or ex3(n,G+) ≥ (1 + o(1))n2, thereby exhibiting a jump for the Turán number of expansions.