Short proofs for long induced paths

TL;DR

This paper introduces a modified depth-first search algorithm to find long induced paths and uses it to establish explicit linear bounds on the induced size-Ramsey number of paths, with improved constants and new proofs for related properties.

Contribution

It provides a simple, modified DFS algorithm for long induced paths and derives explicit linear bounds on the induced size-Ramsey number, improving previous results.

Findings

Induced size-Ramsey number of paths is linearly bounded with explicit constants.

Bound for the k-color version: O(k^3 log^4 k) n.

Supercritical random graphs contain long induced paths of length Θ(ε^2 n).

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 $\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$, 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 $\Theta(\varepsilon^2) n$.