Erdős Problem 917

### What is the conjecture

https://www.erdosproblems.com/917

Let $k\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number).

Is it true that
$$f_k(n) \gg_k n^2?$$
Is it true that
$$f_6(n)\sim n^2/4?$$
More generally, is it true that, for $k\geq 6$,
$$f_k(n) \sim \frac{1}{2}\left(1-\frac{1}{\lfloor k/3\rfloor}\right)n^2?$$

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 conjecture statement and the linked Erdős Problems page for Problem 917. The issue names no repository files, tests, or entry points, so first locate the project’s existing formalized conjecture patterns. Done means adding a formal statement for this conjecture, with any required supporting definitions or proofs identified during the work.

Written by the indexing model from the issue text.