Erdős Problem 626

### What is the conjecture

https://www.erdosproblems.com/626

Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\leq m$). Does
$$\lim_{n\to \infty}\frac{g_k(n)}{\log n}$$
exist?

Conversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does
$$\lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}$$
exist, and what is its value?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Start by reading the Erdős Problems 626 statement at https://www.erdosproblems.com/626 and reviewing formal-conjectures for conventions for adding conjectures. Done means the open conjecture is represented as a Lean formal statement in the repository, with any required definitions and checks passing.

Written by the indexing model from the issue text.