On generalized Tur\'{a}n problems for expansions
Junpeng Zhou, Xiamiao Zhao, Xiying Yuan

TL;DR
This paper extends the study of Turán problems to hypergraph expansions, providing exact and asymptotic results for various classes of expansions, including complete graphs and forests, advancing understanding in extremal hypergraph theory.
Contribution
It systematically investigates generalized Turán numbers for hypergraph expansions, offering new exact and asymptotic results for multiple classes of hypergraphs.
Findings
Exact Turán numbers for expansions of complete graphs.
Asymptotic results for expansions of vertex-disjoint unions of complete graphs.
Asymptotic bounds for expansions of forests like star, linear, and star-path forests.
Abstract
Given a graph , the -expansion of is the -uniform hypergraph obtained from by inserting new distinct vertices in each edge of . Given -uniform hypergraphs and , the generalized Tur\'{a}n number, denoted by , is the maximum number of copies of in an -vertex -uniform hypergraph that does not contain as a subhypergraph. In the case where (i.e., the graph case), the study of generalized Tur\'{a}n problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Tur\'{a}n problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Tur\'{a}n number for expansions…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Markov Chains and Monte Carlo Methods · Stochastic processes and statistical mechanics
