Conjectured by Erdős and Simonovits [ErSi84]. Erdős first offered \$250 for a proof and \$100 for a counterexample, but in [Er93] offered \$500 for a counterexample.

Disproved by Janzer [Ja23b] who constructed, for any $\epsilon>0$, a $3$-regular bipartite graph $H$ such that\[\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}.\]See also [146] and [147] and the entry in the graphs problem collection.

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This page was last edited 19 October 2025.