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On Turán exponents of bipartite graphs

Part of: Graph theory

Published online by Cambridge University Press:  04 August 2021

Tao Jiang ,
Jie Ma  and
Liana Yepremyan
Tao Jiang*
Affiliation:
Department of Mathematics, Miami University, Oxford, OH 45056, USA. Research supported in part by National Science Foundation grants DMS-1400249 and DMS-1855542
Jie Ma
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China. Research supported in part by National Key Research and Development Project SQ2020YFA070080, National Natural Science Foundation of China grants 11501539 and 11622110, and Anhui Initiative in Quantum Information Technologies grant AHY150200
Liana Yepremyan
Affiliation:
Department of Mathematics, University of Oxford, UK. Research supported in part by ERC Consolidator Grant 647678
*
*Corresponding author. Email: jiangt@miamioh.edu.
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Abstract

A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such that $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $k\geq 2$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $k\geq 2$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $^{\prime}$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $^{\prime}$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $^{\prime}$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ .

Keywords

Turán numberTurán exponentextremal functioncube

MSC classification

Primary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 2 , March 2022 , pp. 333 - 344
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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