Ramsey number of 1-subdivisions of transitive tournaments

TL;DR

This paper proves that large enough tournaments contain 1-subdivisions of transitive tournaments, confirming a conjecture and extending classical results from undirected graphs to directed graphs with near-optimal bounds.

Contribution

It establishes the minimum size of tournaments needed to contain 1-subdivisions of transitive tournaments, confirming a conjecture and extending extremal graph theory to directed settings.

Findings

Every tournament with at least (2+o(1))k^2 vertices contains the 1-subdivision of a transitive tournament on k vertices.

The bound is optimal up to a factor of 4.

Confirms a conjecture of Girão, Popielarz, and Snyder.

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 $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ 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 $(2+o(1))k^2$ 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\~ao, Popielarz and Snyder.