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

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	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

Bibliographical note

Funding Information:
Research supported in part by SNSF grant 200021_196965.

Publisher Copyright:
© 2022 The Author(s)

Keywords

Ramsey numbers
Subdivisions
Tournament

ASJC Scopus subject areas

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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AU - Sudakov, Benny

AU - Yuster, Raphael

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Ramsey number of 1-subdivisions of transitive tournaments

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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