# On the growth rate of chromatic numbers of finite subgraphs

## TL;DR

This paper constructs graphs with uncountable chromatic number where all small subgraphs have bounded chromatic number, answering a longstanding question about the growth rate of chromatic numbers in finite subgraphs.

## Contribution

It proves the existence of such graphs for any function, demonstrating a new type of control over the chromatic number growth in infinite graphs.

## Key findings

- Existence of graphs with uncountable chromatic number and controlled finite subgraph chromatic numbers
- Answer to Erdős-Hajnal-Szemerédi question
- Demonstrates the possibility of arbitrarily slow growth of subgraph chromatic numbers

## Abstract

We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices has chromatic number less than $k$. This answers a question of Erd\H{o}s, Hajnal, and Szemeredi.