On Turán exponents of bipartite graphs

<jats:p>A long-standing conjecture of Erdős and Simonovits asserts that for every rational number <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline1. png" /> <jats:tex-math> $r\in (1,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> there exists a bipartite graph H such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline2. png" /> <jats:tex-math> $\mathrm{ex}(n,H)=\Theta(n^r)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. So far this conjecture is known to be true only for rationals of form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline3. png" /> <jats:tex-math> $1+1/k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline4. png" /> <jats:tex-math> $2-1/k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, for integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline5. png" /> <jats:tex-math> $k\geq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we add a new form of rationals for which the conjecture is true: <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline6. png" /> <jats:tex-math> $2-2/(2k+1)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline7. png" /> <jats:tex-math> $k\geq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline8. png" /> <jats:tex-math> $^{\prime}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline9. png" /> <jats:tex-math> $^{\prime}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline10. png" /> <jats:tex-math> $^{\prime}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548321000341_inline11. png" /> <jats:tex-math> $r=7/5$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. </jats:p>