In 2012, Nešetřil and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $Ω(\log \log n)$. In this paper we give an almost matching upper bound by describing, for arbitrarily large values of $n$, 2-degenerate graphs that have a path of order $n$ and where the longest induced paths have order $O((\log \log n)^{1+o(1)})$.

翻译：2012年，Nešetřil与Ossona de Mendez证明了：在有界退化图中，若存在阶为$n$的路径，则必存在阶为$Ω(\log \log n)$的诱导路径。本文通过构造描述给出一个近乎匹配的上界：对于任意大的$n$值，存在2-退化图，其包含阶为$n$的路径，而最长诱导路径的阶仅为$O((\log \log n)^{1+o(1)})$。