AI 中文总结
研究图的极值数有理指数猜想，针对区间中心附近的\(\gamma\)值，即\(\gamma = 1 + \frac{rt - 1}{2rt + 2r}\)（\(r,t\)满足\(t\geq2\)，\(r\geq2t + 3\)），通过特定方法建立了有理指数猜想。
AI 中文摘要
给定一个图\(H\)，极值数\(ex(n,H)\)是\(n\)个顶点的图中不含\(H\)作为子图时的最大边数。著名的埃尔德什和西蒙诺维茨有理指数猜想指出，对于任何有理数\(\gamma\in(1,2)\)，存在一个二分图\(H\)使得\(ex(n,H)=\Theta(n^\gamma)\)。Jiang和Qiu验证了对于所有\(\gamma = 1 + a/b\)（其中\(b > a^2\)）以及Conlon和Janzer验证了对于所有\(\gamma = 2 - a/b\)（其中\(b > \max\{a, (a - 1)^2\}\)）的猜想。本文针对区间中心附近的许多\(\gamma\)，即对于所有\(\gamma = 1 + \frac{rt - 1}{2rt + 2r}\)（其中\(r,t\)是满足\(t\geq2\)，\(r\geq2t + 3\)的自然数）建立了有理指数猜想。
英文摘要
Given a graph $H$, the extremal number $ex(n,H)$ is the maximum number of edges in an $n$-vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjecture of Erdős and Simonovits states that for any rational $γ\in (1,2)$ there exists a single bipartite graph $H$ satisfying $ex(n,H)=Θ(n^γ)$. Among other results, the conjecture has been verified for all $γ=1+a/b$, where $b>a^2$, by Jiang and Qiu and for all $γ=2-a/b$, where $b>\max\{a, (a-1)^2\}$, by Conlon and Janzer. In this paper, we establish the rational exponents conjecture for many $γ$ near the center of the interval, namely, for all $γ=1+\frac{rt-1}{2rt+2r}$, where $r,t$ are natural numbers satisfying $t\geq 2$, $r\geq 2t+3$.
