Long induced paths in $K_{s,s}$-free graphs

More than 40 years ago, Galvin, Rival and Sands showed that every $K_{s,s}$-free graph containing an $n$-vertex path must contain an induced path of length $f(n)$, where $f(n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Recently, it was shown by Duron, Esperet and Raymond that one can take $f(n)=(\log \log n{)}^{1\mathrm{/}5-o(1)}$. In this note, we give a short self-contained proof that a $K_{s,s}$-free graphs with an $n$-vertex path contains an induced path of length at least $(\log \log n{)}^{1-o(1)}$. Combined with the recent remarkable example of Couëtoux, Defrain, and Raymond, which provides an upper bound of $O((\log \log n{)}^{1+o(1)})$, this essentially resolves this old problem.