A survey of Turán problems for expansions

2015/05/29 by Mubayi, Dhruv, Verstraete, Jacques · 3 citations
#05-xx #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1505.08078

Abstract

The r-expansion G⁺ of a graph G is the r-uniform 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 ex_r(n,F) denote the maximum number of edges in an r-uniform hypergraph with n vertices not containing any copy of the r-uniform hypergraph F. Many problems in extremal set theory ask for the determination of ex_r(n,G⁺) for various graphs G. We survey these Turán-type problems, focusing on recent developments.

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• Spectral Turán-type problems for the α-spectral radius of hypergraphs with degree stability
• A generalised Ramsey--Turán problem for matchings
• The maximum number of triangles in K_1,s,t-free graphs

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