小群的有理指数

Sean English, Anastasia Halfpap, Robert A. Krueger
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引用次数: 0

摘要

让 $\mathrm{ex}(n,H,\mathcal{F})$ 是一个 $n$ 顶点图中 $H$ 的最大副本数,这个图不包含来自 $\mathcal{F}$ 的图的副本。把 $H$ 和 $\mathcal{F}$ 看作是固定的,我们研究在 $n$ 中 $\mathrm{ex}(n,H,\mathcal{F})$ 的渐近性。如果存在一个有限族$\mathcal{F}$,使得$mathrm{ex}(n,H,\mathcal{F}) = \Theta(n^r)$,我们就说有理数$r$对于$H$是可实现的。布克和康伦利用随机代数构造证明,介于 1$ 与 2$ 之间的每一个有理数对于 $K_2$ 都是可实现的。我们将他们的结果推广到表明,对于所有 $t \geq 2$,每一个介于$1$和$t$之间的有理数对于 $K_t$ 都是可实现的。我们还确定了恒星的可变现有理数,并指出了与相关的西多伦科型超饱和问题的联系。
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Rational exponents for cliques
Let $\mathrm{ex}(n,H,\mathcal{F})$ be the maximum number of copies of $H$ in an $n$-vertex graph which contains no copy of a graph from $\mathcal{F}$. Thinking of $H$ and $\mathcal{F}$ as fixed, we study the asymptotics of $\mathrm{ex}(n,H,\mathcal{F})$ in $n$. We say that a rational number $r$ is \emph{realizable for $H$} if there exists a finite family $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F}) = \Theta(n^r)$. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between $1$ and $2$ is realizable for $K_2$. We generalize their result to show that every rational between $1$ and $t$ is realizable for $K_t$, for all $t \geq 2$. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem.
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