Induced trees in triangle-free graphs

TL;DR

This paper proves that every connected triangle-free graph contains a large induced tree of exponential size in the square root of log n, advancing understanding of graph structure.

Contribution

It establishes a lower bound on the size of induced trees in triangle-free graphs, improving previous bounds and addressing longstanding questions.

Findings

Every connected triangle-free graph on n vertices has an induced tree of size at least exp(c√log n).

The result partially answers open questions by Erdos, Saks, Sos, and Pultr.

Provides a new lower bound that approaches the known upper bound.

Abstract

We prove that every connected triangle-free graph on $n$ vertices contains an induced tree on $\exp(c\sqrt{\log n})$ vertices, where $c$ is a positive constant. The best known upper bound is $(2+o(1))\sqrt n$. This partially answers questions of Erdos, Saks, and Sos and of Pultr.