Erd\H{o}s, Fajtlowicz and Staton asked for the least integer $f(k)$ such that every graph with more than $f(k)$ vertices has an induced regular subgraph with at least $k$ vertices. Here we consider the following relaxed notions. Let $g(k)$ be the least integer such that every graph with more than $g(k)$ vertices has an induced subgraph with at least $k$ repeated degrees and let $h(k)$ be the least integer such that every graph with more than $h(k)$ vertices has an induced subgraph with at least $k$ maximum degree vertices. We obtain polynomial lower bounds for $h(k)$ and $g(k)$ and nontrivial linear upper bounds when the host graph has bounded maximum degree.

中文翻译：

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具有许多重复度数的诱导子图

Erd\H{o}s、Fajtlowicz 和 Staton 要求最小整数 $f(k)$，使得每个具有超过 $f(k)$ 个顶点的图都有一个包含至少 $k$ 个顶点的诱导正则子图。在这里，我们考虑以下宽松的概念。令 $g(k)$ 是最小整数，使得每个具有超过 $g(k)$ 个顶点的图都有一个至少有 $k$ 个重复度数的诱导子图，并让 $h(k)$ 是最小整数，例如每个具有超过 $h(k)$ 个顶点的图都有一个包含至少 $k$ 个最大度顶点的诱导子图。我们获得了 $h(k)$ 和 $g(k)$ 的多项式下界以及当宿主图具有最大度有界时的非平凡线性上界。