If $G$ is a graph with chromatic number $\chi(G)=m$ then must $G$ contain a subgraph $H$ with \[\zeta(H) \gg \frac{m}{\log m}?\]
If $G$ is a graph with chromatic number $\chi(G)=m$ then must $G$ contain a subgraph $H$ with \[\zeta(H) \gg \frac{m}{\log m}?\]
The answer is yes, proved by Alon, Krivelevich, and Sudakov [AKS97].