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On asymptotic local Turán problems

Peter Frankl and Jiaxi Nie

Vol. 12 (2023), No. 4, 273–286
DOI: 10.2140/moscow.2023.12.273
Abstract

An r-uniform hypergraph has the (q,p)-property if any set of q vertices spans a complete subhypergraph on p vertices. Let tr(n,q,p) be the minimum edge density of an n-vertex r-uniform hypergraph with the (q,p)-property and let tr(q,p) = lim ntr(n,q,p). A disjoint union of k complete hypergraphs has the (q,qk)-property, which gives tr((q,qk)) 1kr1 . The first author, Huang and Rödl showed that these constructions are the best asymptotically, that is, lim qtr((q,qk)) = 1kr1 . They asked whether it is true for all real numbers γ 1 that lim qtr((q,qγ)) = 1γr1 . We give positive answers to this question for a small range of real numbers, and, on the other hand, provide new constructions that give negative answers for many other ranges.

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Keywords
Turán problem, hypergraph, extremal combinatorics
Mathematical Subject Classification
Primary: 05D05
Milestones
Received: 1 March 2023
Revised: 25 July 2023
Accepted: 26 August 2023
Published: 8 December 2023
Authors
Peter Frankl
Rényi Institute
Hungarian Academy of Sciences
Budapest
Hungary
Jiaxi Nie
Shanghai Center for Mathematical Sciences
Fudan University
Shanghai
China