Short proofs for long induced paths

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