A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such that $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $k\geq 2$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $k\geq 2$ . 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 $^{\prime}$ 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 $^{\prime}$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $^{\prime}$ 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: $r=7/5$ .

Erdős 和 Simonovits 的一个长期猜想断言对于每个有理数 $r\in (1,2)$ 都存在一个二分图 H 使得 $\mathrm{ex}(n,H)=\Theta(n^ r)$ 。到目前为止，这个猜想仅对形式 为 $1+1/k$ 和 $2-1/k$ 的有理数是正确的，对于整数 $k\geq 2$ 。在本文中，我们添加了一种新形式的有理数，其猜想为真： $2-2/(2k+1)$ ，对于 $k\geq 2$ 。这反过来也对 Pinchasi 和 Sharir 关于立方体图的问题给出了肯定的回答。最近，一个版本的 Erdős 和 Simonovits $^{\prime}$ Bukh 和 Conlon 证实了用有限族替换单个图的猜想。他们提出了一个二分图的构造，它应该满足 Erdős 和 Simonovits $^{\prime}$ 的猜想。 我们的结果也可以看作是验证 Bukh 和 Conlon $^{\prime}$ 猜想的第一步。我们还证明了不对称设置中 theta 图的 Turán 数的上限，并利用该结果为 Turán 指数获得另一个新的有理指数： $r=7/5$ 。