A quasi-optimal upper bound for induced paths in sparse graphs

In 2012, Ne\v{s}et\v{r}il and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $\Omega(\log \log n)$. In this paper we give an almost matching upper bound by describing, for arbitrarily large values of $n$, a 2-degenerate graph that has a path of order $n$ and where all induced paths have order $O(\log \log n \cdot \log \log \log n)$.