  Graphs of bounded degeneracy are known to contain induced paths of order
$\Omega(\log \log n)$ when they contain a path of order $n$, as proved by
Ne\v{s}et\v{r}il and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and
Maffray conjectured that this bound could be improved to $\Omega((\log n)^c)$
for some constant $c>0$ depending on the degeneracy. We disprove this
conjecture by constructing, for arbitrarily large values of $n$, a graph that
is 2-degenerate, has a path of order $n$, and where all induced paths have
order $O((\log \log n)^2)$. We also show that the graphs we construct have
linearly bounded coloring numbers.

Combinatorics;Discrete Mathematics