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Many Turán exponents via subdivisions

Part of: Graph theory

Published online by Cambridge University Press:  21 July 2022

Tao Jiang  and
Yu Qiu
Tao Jiang*
Affiliation:
Department of Mathematics, Miami University, Oxford, OH 45056, USA
Yu Qiu
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China
*
*Corresponding author. Email: jiangt@miamioh.edu
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Abstract

Given a graph $H$ and a positive integer $n$, the Turán number $\mathrm{ex}(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in (1,2)$ is called a Turán exponent if there exists a bipartite graph $H$ such that $\mathrm{ex}(n,H)=\Theta (n^r)$. A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q\gt p$.

In this paper, we show that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q \gt p^{2}$. Our result also addresses a conjecture of Janzer [18].

Keywords

rational exponent conjectureTurán numberssubdivisions

MSC classification

Primary: 05C35: Extremal problems

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 1 , January 2023 , pp. 134 - 150
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

†

Research supported in part by NSF grant DMS-1855542.

‡

Research supported in part by China Scholarship Council grant #201806340156.

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