诱导子图的绝对可避免阶数对
Absolutely avoidable order-size pairs for induced subgraphs
摘要：
我们称一个整数对$(m,f)$为$m\geq 1$，$0\leq f \leq \binom{m}{2}$，\emph{absolutely avoidable}，如果存在$n_0$使得对于任何一个具有$n>n_0$和$0\leq e\leq \binom{n}{2}$的整数对$(n,e)$，都存在一个在$n$顶点和$e$边上的图，并且在$m$顶点和$f$边上不包含导出子图.一些对显然不是绝对可避免的，例如$(m,0)$不是绝对可避免的，因为任何至少在$m$顶点上的足够稀疏的图都包含在$m$顶点上的独立集。在这里，我们表明，有无穷多个绝对可避免的对。我们给出了一个特定的无限集$M$，使得对于任何$m\in M$，对$(m, \binom{m}{2}/2)$是绝对可避免的。此外，在其他结果中，我们表明，对于任何单调整数函数$q(m)$，$|q(m)|=O(m)$，有无穷多个值的$m$，使对$(m, \binom{m}{2}/2 +q(m))$是绝对可避免的。
英文摘要：
We call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{2}$, \emph{absolutely avoidable} if there is $n_0$ such that for any pair of integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{2}$ there is a graph on $n$ vertices and $e$ edges that contains no induced subgraph on $m$ vertices and $f$ edges. Some pairs are clearly not absolutely avoidable, for example $(m,0)$ is not absolutely avoidable since any sufficiently sparse graph on at least $m$ vertices contains independent sets on $m$ vertices. Here we show that there are infinitely many absolutely avoidable pairs. We give a specific infinite set $M$ such that for any $m\in M$, the pair $(m, \binom{m}{2}/2)$ is absolutely avoidable. In addition, among other results, we show that for any monotone integer function $q(m)$, $|q(m)|=O(m)$, there are infinitely many values of $m$ such that the pair $(m, \binom{m}{2}/2 +q(m))$ is absolutely avoidable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
