The extremal number $\mathrm{ex}(n,F)$ of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number $r\in [1,2]$ is realisable if there exists a graph F with $\mathrm{ex}(n,F)=\mathrm{\Theta }(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1,\frac{7}{5},2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m\ge 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than the two numbers 1 and 2.

In this paper, we make progress on the conjecture of Erdős and Simonovits. First, we show that $2-\frac{a}{b}$ is realisable for any integers $a,b\ge 1$ with $b>a$ and $b\equiv \pm 1(\mathrm{mod}a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence.

Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

极数 $\mathrm{前}（ñ，F）$图F的“ F”是不包含F作为子图的n-顶点图中最大边数。实数$\mathrm{[R}\in [\mathrm{1个}，2]$是可实现的，如果存在的曲线图˚F与$\mathrm{前}（ñ，F）=\mathrm{\Theta }（{ñ}^{\mathrm{[R}}）$。几十年前，Erdős和Simonovits推测，$[\mathrm{1个}，2]$是可以实现的。尽管经过数十年的努力，唯一已知的可实现数字是$0，\mathrm{1个}，\frac{7}{5}，2$，以及表格的编号 $\mathrm{1个}+\frac{\mathrm{1个}}{米}$， $2-\frac{\mathrm{1个}}{米}$， $2-\frac{2}{米}$ 对于整数 $米\ge \mathrm{1个}$。尤其是，甚至不知道所有可实现数字的集合是否包含两个数字1和2以外的单个极限点。

在本文中，我们在Erdős和Simonovits的猜想上取得了进展。首先，我们证明$2-\frac{\mathrm{一种}}{b}$ 对于任何整数均可实现 $\mathrm{一种}，b\ge \mathrm{1个}$ 与 $b>\mathrm{一种}$ 和 $b\equiv \pm \mathrm{1个}（\mathrm{模}\mathrm{一种}）$。这包括所有先前已知的极限，并给出了无限多个极限点$2-\frac{\mathrm{1个}}{米}$ 结果是所有可实现数字的集合中。

其次，我们对二部图的细分提出了一个猜想。除了令人感兴趣之外，我们还显示出令人惊讶的是，这种细分猜想实际上暗示着介于1和2之间的每个有理数都是可以实现的。