Oriented Ramsey numbers of graded digraphs

TL;DR

This paper establishes that graded digraphs with bounded maximum degree have linear upper bounds on their oriented Ramsey numbers, and demonstrates the near-optimality of these bounds through matching lower bounds.

Contribution

It proves a linear upper bound on the oriented Ramsey number for graded digraphs with bounded degree and shows this bound is essentially tight.

Findings

Oriented grids in fixed dimensions have linear Ramsey numbers.

Polynomial bounds are established for hypercube Ramsey numbers.

The bounds are shown to be nearly optimal with matching lower bounds.

Abstract

We show that any graded digraph $D$ on $n$ vertices with maximum degree $\Delta$ has an oriented Ramsey number of at most $C^\Delta n$ for some absolute constant $C > 1$, improving upon a recent result of Fox, He, and Wigderson. In particular, this implies that oriented grids in any fixed dimension have linear oriented Ramsey numbers, and gives a polynomial bound on the oriented Ramsey number of the hypercube. We also show that this result is essentially best possible, in that there exist graded digraphs on $n$ vertices with maximum degree $\Delta$ such that their oriented Ramsey number is at least $c^\Delta n$ for some absolute constant $c > 1$.