If $H$ is bipartite and $r$-degenerate, is $\mathrm{ex}(n;H) \ll n^{2-1/r}$ (a $500 Erdős-Simonovits prize conjecture)? A counterexample refutes the degeneracy conjecture.

If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $$\mathrm{ex}(n;H) \ll n^{2-1/r}.$$

The answer is no. OpenAI [OpenAI26] give a connected bipartite 2-degenerate H and constants c, ε > 0 with $\mathrm{ex}(n;H)\geq cn^{3/2+\epsilon}$ for all large n, which exceeds the conjectured $n^{2-1/2}=n^{3/2}$. See erdos_146.variants.two_degenerate_counterexample.