Problem #571: Show that for any rational $\alpha \in [1,2)$ there exists...

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Progress

A problem of Erdős and Simonovits.

Bukh and Conlon [BuCo18] proved that this holds if we weaken asking for the extremal number of a single graph to asking for the extremal number of a finite family of graphs.

A rational $\alpha\in [1,2)$ for which this holds is known as a Turán exponent. Known Turán exponents are:

• $\frac{3}{2}-\frac{1}{2s}$ for $s\geq 2$ (Conlon, Janzer, and Lee [CJL21]).


• $\frac{4}{3}-\frac{1}{3s}$ and $\frac{5}{4}-\frac{1}{4s}$ for $s\geq 2$ (Jiang and Qiu [JiQi20]).


• $2-\frac{a}{b}$ for $\lfloor b/a\rfloor^3 \leq a\leq \frac{b}{\lfloor b/a\rfloor+1}+1$ (Jiang, Jiang, and Ma [JJM20]).


• $2-\frac{a}{b}$ with $b>a\geq 1$ and $b\equiv \pm 1\pmod{a}$ (Kang, Kim, and Liu [KKL21]).


• $1+a/b$ with $b>a^2$ (Jiang and Qiu [JiQi23]),


• $2-\frac{2}{2b+1}$ for $b\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan [JMY22]).


• $2-a/b$ with $b\geq (a-1)^2$ (Conlon and Janzer [CoJa22]).

See also [713] and the entry in the graphs problem collection.