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Mat. Zametki, 2014, Volume 95, Issue 2, Pages 271–281 (Mi mz10187)  

This article is cited in 2 scientific papers (total in 2 papers)

On Turán's $(3,4)$-Problem with Forbidden Subgraphs

A. A. Razborovab

a Steklov Mathematical Institute of the Russian Academy of Sciences
b University of Chicago, USA

Abstract: We identify three $3$-graphs on five vertices that are missing in all known extremal configurations for Turán's $(3,4)$-problem and prove Turán's conjecture for $3$-graphs that are additionally known not to contain any induced copies of these $3$-graphs. Our argument is based on an (apparently) new technique of “indirect interpretation” that allows us to retrieve additional structure from hypothetical counterexamples to Turán's conjecture, but in rather loose and limited sense. We also include two miscellaneous calculations in flag algebras that prove similar results about some other additional forbidden subgraphs.

Keywords: Turán's $(3,4)$-problem, $3$-graph, hypergraph, forbidden subgraph.

Funding Agency Grant Number
Russian Foundation for Basic Research


DOI: https://doi.org/10.4213/mzm10187

Full text: PDF file (485 kB)
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English version:
Mathematical Notes, 2014, 95:2, 247–254

Bibliographic databases:

UDC: 519.113
Received: 13.12.2012
Revised: 23.03.2013

Citation: A. A. Razborov, “On Turán's $(3,4)$-Problem with Forbidden Subgraphs”, Mat. Zametki, 95:2 (2014), 271–281; Math. Notes, 95:2 (2014), 247–254

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Norin S., Yepremyan L., “Sparse Halves in Dense Triangle-Free Graphs”, J. Comb. Theory Ser. B, 115 (2015), 1–25  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. Raymond, J. Saunderson, M. Singh, R. R. Thomas, “Symmetric sums of squares over $k$-subset hypercubes”, Math. Program., 167:2 (2018), 315–354  crossref  mathscinet  zmath  isi  scopus
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