Ramsey numbers of digraphs with local edge structure
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911139462381568 |
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| author | Bradač, Domagoj Morawski, Patryk Sudakov, Benny Wigderson, Yuval |
| author_facet | Bradač, Domagoj Morawski, Patryk Sudakov, Benny Wigderson, Yuval |
| contents | One of the classical topics in graph Ramsey theory is the study of which $n$-vertex graphs have Ramsey numbers that are linear in $n$. In this paper, we consider this problem in the context of directed graphs. The oriented Ramsey number of a digraph $G$ is the smallest integer $N$ such that every $N$-vertex tournament contains a copy of $G$. We prove that every bounded-degree acyclic digraph with a ``local edge structure'' has a linear oriented Ramsey number.
More precisely, we say that a digraph $G$ has graded bandwidth $w$ if its vertices can be partitioned into sets $V_1, \dots, V_H$ such that all edges $uv \in E(G)$ with $u \in V_i$ and $v \in V_j$ satisfy $1 \leq j - i \leq w$. We prove that $\vec{r}(G) \leq 3^{57Δw} |V(G)|$ for any acyclic $G$ with graded bandwidth $w$ and maximum degree $Δ$.
This provides a common generalization of several prior results, including on digraphs of bounded height, of digraphs of bounded bandwidth, and blowups of bounded-degree oriented trees. This notion also captures a wide variety of natural digraphs, such as oriented grids and hypercubes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05055 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ramsey numbers of digraphs with local edge structure Bradač, Domagoj Morawski, Patryk Sudakov, Benny Wigderson, Yuval Combinatorics One of the classical topics in graph Ramsey theory is the study of which $n$-vertex graphs have Ramsey numbers that are linear in $n$. In this paper, we consider this problem in the context of directed graphs. The oriented Ramsey number of a digraph $G$ is the smallest integer $N$ such that every $N$-vertex tournament contains a copy of $G$. We prove that every bounded-degree acyclic digraph with a ``local edge structure'' has a linear oriented Ramsey number. More precisely, we say that a digraph $G$ has graded bandwidth $w$ if its vertices can be partitioned into sets $V_1, \dots, V_H$ such that all edges $uv \in E(G)$ with $u \in V_i$ and $v \in V_j$ satisfy $1 \leq j - i \leq w$. We prove that $\vec{r}(G) \leq 3^{57Δw} |V(G)|$ for any acyclic $G$ with graded bandwidth $w$ and maximum degree $Δ$. This provides a common generalization of several prior results, including on digraphs of bounded height, of digraphs of bounded bandwidth, and blowups of bounded-degree oriented trees. This notion also captures a wide variety of natural digraphs, such as oriented grids and hypercubes. |
| title | Ramsey numbers of digraphs with local edge structure |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.05055 |