Ramsey numbers of sparse digraphs

TL;DR

This paper investigates the Ramsey numbers of bounded-degree acyclic digraphs, disproving a natural conjecture by showing they can grow super-polynomially, but also establishing nearly linear bounds for typical cases and several families.

Contribution

It disproves the conjecture that the oriented Ramsey number is always linear for bounded-degree acyclic digraphs, and provides bounds for various cases including multiple colors.

Findings

Existence of acyclic digraphs with super-polynomial Ramsey numbers

Nearly linear bounds for typical bounded-degree acyclic digraphs

Quasi-polynomial upper bounds for multi-color Ramsey numbers

Abstract

Burr and Erd\H{o}s in 1975 conjectured, and Chv\'atal, R\"odl, Szemer\'edi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erd\H{o}s conjecture, answering a question of Buci\'c, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $\Delta \geq 2$ and any sufficiently large $n$, there exists an acyclic digraph $H$ with $n$ vertices and maximum degree $\Delta$ such that \[ \overrightarrow{r_{1}}(H)\ge n^{\Omega(\Delta^{2/3}/ \log^{5/3} \Delta)}. \] This proves that $\overrightarrow{r_{1}}(H)$ is not always linear in the number of vertices for bounded-degree $H$. On the other hand, we show that $\overrightarrow{r_{1}}(H)$ is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs $H$, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs. For multiple colors, we prove a quasi-polynomial upper bound $\overrightarrow{r_{k}}(H)=2^{(\log n)^{O_{k}(1)}}$ for all bounded-degree acyclic digraphs $H$ on $n$ vertices, where $\overrightarrow{r_k}(H)$ is the least $N$ such that every $k$-edge-colored tournament on $N$ vertices contains a monochromatic copy of $H$. For $k\geq 2$ and $n\geq 4$, we exhibit an acyclic digraph $H$ with $n$ vertices and maximum degree $3$ such that $\overrightarrow{r_{k}}(H)\ge n^{\Omega(\log n/\log\log n)}$, showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.