Many Turán exponents via subdivisions

<jats:p>Given a graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline1. png" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and a positive integer <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline2. png" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the <jats:italic>Turán number</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline3. png" /> <jats:tex-math> $\mathrm{ex}(n,H)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximum number of edges in an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline4. png" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-vertex graph that does not contain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline5. png" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as a subgraph. A real number <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline6. png" /> <jats:tex-math> $r\in (1,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is called a <jats:italic>Turán exponent</jats:italic> if there exists a bipartite graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline7. png" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline8. png" /> <jats:tex-math> $\mathrm{ex}(n,H)=\Theta (n^r)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. A long-standing conjecture of Erdős and Simonovits states that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline9. png" /> <jats:tex-math> $1+\frac{p}{q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a Turán exponent for all positive integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline10. png" /> <jats:tex-math> $p$ </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="S0963548322000177_inline11. png" /> <jats:tex-math> $q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline12. png" /> <jats:tex-math> $q\gt p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. </jats:p> <jats:p>In this paper, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline13. png" /> <jats:tex-math> $1+\frac{p}{q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a Turán exponent for all positive integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline14. png" /> <jats:tex-math> $p$ </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="S0963548322000177_inline15. png" /> <jats:tex-math> $q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline16. png" /> <jats:tex-math> $q \gt p^{2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our result also addresses a conjecture of Janzer [18]. </jats:p>