Conlon, David and Janzer, Oliver and Lee, Joonkyung
(2019)
*More on the extremal number of subdivisions.*
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(Unpublished)
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170943053

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## Abstract

Given a graph H, the extremal number ex(n,H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing K'_(s,t) for the subdivision of the bipartite graph K_(s,t), we show that ex(n,K'_(s,t)) = O(n^((3/2) - 1/(2s))). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s, k ≥ 1, we show that ex(n, L) = Θ(n^(1 + s/(sk+1))) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ϵ (1,2) is realisable in the sense that ex(n,H) = Θ(n^r) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing H^k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n,H^(k-1)) = O(n^(1 + 1/k - δ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C_4 as a subgraph satisfies ex(n, H) = o(n^(2 - 1/r).

Item Type: | Report or Paper (Discussion Paper) | ||||||
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Additional Information: | Conlon research supported by ERC Starting Grant RanDM 676632. Lee research supported by ERC Consolidator Grant PEPCo 724903. | ||||||

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Record Number: | CaltechAUTHORS:20190819-170943053 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20190819-170943053 | ||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 98035 | ||||||

Collection: | CaltechAUTHORS | ||||||

Deposited By: | Melissa Ray | ||||||

Deposited On: | 20 Aug 2019 23:02 | ||||||

Last Modified: | 03 Oct 2019 21:37 |

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