关于广义有理指数猜想
On the Generalized Rational Exponents Conjecture
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中文总结 AI 辅助
本文证明了Gerbner与Palmer提出的广义有理指数猜想，通过局部化-压缩-移位框架构造出直径≤3的连通图H_α与F_α，使得对应广义Turán数为Θ(n^α)。
中文摘要 AI 辅助
对于固定图H和F，令ex(n,H,F)表示n顶点不含F的图中H的副本的最大数量。本文证明了由Gerbner和Palmer提出的广义有理指数猜想，即对于每个有理数α≥1，存在固定图H_α和F_α，使得ex(n,H_α,F_α)=Θ(n^α)。此外，计数图H_α总能被选为直径不超过3的连通图。我们的论证依赖于局部化-压缩-移位框架，该框架将Bukh-Conlon针对边的有限族构造转化为含单个禁图的广义Turán问题场景。
英文摘要
For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $α\ge1$, there exist fixed graphs $H_α$ and $F_α$ such that \[ \ex(n,H_α,F_α)=Θ(n^α). \] Furthermore, the counting graph $H_α$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization--compression--shift framework, which transforms the Bukh--Conlon finite family construction for edges into a generalized Turán problem setting with a single forbidden graph.
