Problem 571

Erdős Problem #571: declared status 'open'. Formalized: no. Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that$$\mathrm{ex}(n;G)\asymp n^{\alpha}.$$ Current best: Known Tur\'{a}n exponents are: {UL} {LI} $\frac{3}{2}-\frac{1}{2s}$ for $s\geq 2$ (Conlon, Janzer, and Lee [CJL21]).{/LI} {LI} $\frac{4}{3}-\frac{1}{3s}$ and $\frac{5}{4}-\frac{1}{4s}$ for $s\geq 2$ (Jiang and Qiu [JiQi20]).{/LI} {LI} $2-\frac{a}{b}$ for $\lfloor b/a\rfloor^3 \leq a\leq \frac{b}{\lfloor b/a\rfloor+1}+1$ (Jiang, Jiang, and Ma [JJM20]).{/LI} {LI} $2-\frac{a}{b}$ with $b>a\geq 1$ and $b\equiv \pm 1\pmod{a}$ (Kang, Kim, and Liu [KKL21]).{/LI} {LI} $1+a/b$ with $b>a^2$ (Jiang and Qiu [JiQi23]),{/LI} {LI} $2-\frac{2}{2b+1}$ for $b\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan [JMY22]).{/LI} {LI} $2-a/b$ with $b\geq (a-1)^2$ (Conlon and Janzer [CoJa22]).{/LI} {/UL} See also [713] and the entry in the graphs problem collection. Prize: no. Tags: graph theory, turan number.