  We present a modification of the Depth first search algorithm, suited for
finding long induced paths. We use it to give simple proofs of the following
results. We show that the induced size-Ramsey number of paths satisfies
$\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$, thus giving an explicit
constant in the linear bound, improving the previous bound with a large
constant from a regularity lemma argument by Haxell, Kohayakawa and {\L}uczak.
We also provide a bound for the $k$-color version, showing that
$\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$. Finally, we present a new short
proof of the fact that the binomial random graph in the supercritical regime,
$G(n,\frac{1+\varepsilon}{n})$, contains typically an induced path of length
$\Theta(\varepsilon^2) n$.
