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

TL;DR

This paper proves that in $K_{s, s}$-free graphs with an $n$-vertex path, there exists an induced path of length approximately $( ext{log log } n)^{1-o(1)}$, resolving a long-standing problem in graph theory.

Contribution

The authors provide a concise, self-contained proof establishing a lower bound on the length of induced paths in $K_{s, s}$-free graphs, matching recent upper bounds.

Findings

Induced path length in $K_{s,s}$-free graphs is at least $( ext{log log } n)^{1-o(1)}$.

This result essentially resolves a 40-year-old open problem.

The proof aligns with recent upper bounds, confirming the tightness of the bounds.

Abstract

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 $f(n)\to \infty$ as $n\to \infty$. Recently, it was shown by Duron, Esperet and Raymond that one can take $f(n)=(\log \log n)^{1/5-o(1)}$. 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 Cou\"etoux, Defrain, and Raymond, which provides an upper bound of $O((\log \log n)^{1+o(1)})$, this essentially resolves this old problem.