The near-linear-order Hajós conjecture for critical graphs

Let $G$ be an $r$-critical graph on $r+o(r)$ vertices, where an $r$-critical graph has chromatic number $r$ and every proper subgraph has smaller chromatic number. A graph satisfies the Hajós conjecture when it contains a subdivision of $K_r$. Near-linear-order Hajós conjecture. $G$ satisfies the Hajós conjecture.

The paper has established the assertion for $r$-critical graphs on at most $r+4$ vertices, and proposes extending those methods to the broader near-linear-order regime; the general statement remains open.