Download this article
 Download this article For screen
For printing
Recent Issues
Volume 13, Issue 1
Volume 12, Issue 4
Volume 12, Issue 3
Volume 12, Issue 2
Volume 12, Issue 1
Volume 11, Issue 4
Volume 11, Issue 3
Volume 11, Issue 2
Volume 11, Issue 1
Volume 10, Issue 4
Volume 10, Issue 3
Volume 10, Issue 2
Volume 10, Issue 1
Volume 9, Issue 4
Volume 9, Issue 3
Volume 9, Issue 2
Volume 9, Issue 1
Volume 8, Issue 4
Volume 8, Issue 3
Volume 8, Issue 2
Volume 8, Issue 1
Older Issues
Volume 7, Issue 4
Volume 7, Issue 3
Volume 7, Issue 2
Volume 7, Issue 1
Volume 6, Issue 4
Volume 6, Issue 2-3
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 1-2
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 3-4
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN (electronic): 2996-220X
ISSN (print): 2996-2196
Author Index
To Appear
 
Other MSP Journals
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.

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