Ramsey number of 1-subdivisions of transitive tournaments

Nemanja Draganić; David Munhá Correia; Benny Sudakov; Raphael Yuster

doi:10.1016/j.jctb.2022.06.001

Ramsey number of 1-subdivisions of transitive tournaments

Nemanja Draganić, David Munhá Correia, Benny Sudakov, Raphael Yuster

Department of Mathematics

University of Haifa

Research output: Contribution to journal › Article › peer-review

Abstract

The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every n-vertex graph with at least εn² edges contains a 1-subdivision of the complete graph on c_εn vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least (2+o(1))k² vertices contains the 1-subdivision of a transitive tournament on k vertices. This is optimal up to a multiplicative factor of 4 and confirms a conjecture of Girão, Popielarz and Snyder.

Original language	American English
Pages (from-to)	176-183
Number of pages	8
Journal	Journal of Combinatorial Theory. Series B
Volume	157
DOIs	https://doi.org/10.1016/j.jctb.2022.06.001
State	Published - Nov 2022

Keywords

Ramsey numbers
Subdivisions
Tournament

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jctb.2022.06.001

Cite this

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title = "Ramsey number of 1-subdivisions of transitive tournaments",

abstract = "The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erd{\H o}s, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every n-vertex graph with at least εn2 edges contains a 1-subdivision of the complete graph on cεn vertices, resolving another old conjecture of Erd{\H o}s. In this paper we consider the directed analogue of these problems and show that every tournament on at least (2+o(1))k2 vertices contains the 1-subdivision of a transitive tournament on k vertices. This is optimal up to a multiplicative factor of 4 and confirms a conjecture of Gir{\~a}o, Popielarz and Snyder.",

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year = "2022",

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doi = "10.1016/j.jctb.2022.06.001",

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T1 - Ramsey number of 1-subdivisions of transitive tournaments

AU - Draganić, Nemanja

AU - Munhá Correia, David

AU - Sudakov, Benny

AU - Yuster, Raphael

PY - 2022/11

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AB - The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every n-vertex graph with at least εn2 edges contains a 1-subdivision of the complete graph on cεn vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least (2+o(1))k2 vertices contains the 1-subdivision of a transitive tournament on k vertices. This is optimal up to a multiplicative factor of 4 and confirms a conjecture of Girão, Popielarz and Snyder.

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DO - 10.1016/j.jctb.2022.06.001

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JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -

Ramsey number of 1-subdivisions of transitive tournaments

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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