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.
Bibliographical noteFunding Information:
Research supported in part by SNSF grant 200021_196965.
© 2022 The Author(s)
- Ramsey numbers
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics