Short proofs for long induced paths

Abstract

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 ${\hat{R}}_{\mathrm{ind}}(P_n)\le 5\cdot 10^{7}n$, 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 Łuczak. We also provide a bound for the $k$-color version, showing that ${\hat{R}}_{\mathrm{ind}}^k(P_n)=O(k^3{\log }^{4}k)n$. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,\frac{1+\epsilon }{n})$, contains typically an induced path of length $\Theta ({\epsilon }^2)n$.

arXiv commentto appear in CPC