# List Ramsey numbers

## TL;DR

This paper extends classical Ramsey numbers to list colouring, exploring their relationship, bounds, and differences, especially for uniform cliques, revealing new bounds and cases of equality and disparity.

## Contribution

It introduces list Ramsey numbers, analyzes their relation to classical numbers, and establishes exponential bounds for uniform cliques, contrasting with known super-exponential growth.

## Key findings

- List Ramsey numbers can be equal or far apart from classical Ramsey numbers.
- For $oldsymbol{ ext{l}}$-uniform cliques, list Ramsey numbers are exponentially bounded.
- Classical Ramsey numbers for uniform cliques grow super-exponentially.

## Abstract

We introduce the list colouring extension of classical Ramsey numbers. We investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For $\ell$-uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well-known that the Ramsey number is super-exponential for uniformity at least $3$. This is in great contrast to the graph case where we cannot even decide the question of equality for cliques.