Erdős Problem #147

If \(H\) is bipartite with minimum degree \(r\) then there exists \(\epsilon=\epsilon(H)>0\) such that \[\mathrm{ex}(n;H) \gg n^{2-\frac{1}{r-1}+\epsilon}.\] Conjectured by Erdős and Simonovits [ErSi84]. A probabilistic argument shows that there exists some \(\epsilon=\epsilon(H)>0\) such that \(\mathrm{ex}(n;H) \gg n^{2-\frac{2}{r}+\epsilon}\). This conjecture was disproved by Janzer [Ja23] for even \(r\geq 4\). The case \(r=3\) was disproved by Janzer [Ja23b], who constructed, for any \(\epsilon>0\), a \(3\)-regular bipartite graph \(H\) such that \(\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}\). In [Ja23] Janzer conjectures that the above lower bound is sharp, in that for any \(r\geq 3\) and \(\epsilon>0\) there exists an \(r\)-regular graph \(H\) such that \(\mathrm{ex}(n;H) \ll n^{2-\frac{2}{r}+\epsilon}\). Janzer's result proves this for even \(r\geq 4\). See also [113](https://www.erdosproblems.com/113), [146](https://www.erdosproblems.com/146), and [714](https://www.erdosproblems.com/714). The conjecture is stated for minimum degree \(r \geq 2\) (for \(r = 1\) the exponent \(2 - \frac{1}{r-1}\) is not meaningful).