A phase transition in Erd\H{o}s-Barak random graphs

TL;DR

This paper refines the understanding of the phase transition for monotone paths in Erdős-Rényi graphs, pinpointing the critical probability and window with greater precision.

Contribution

It improves the known critical value for the phase transition and identifies the precise critical window in Erdős-Rényi random graphs.

Findings

Critical probability refined to (log n - log log n)/n

Identified the critical window of order Θ(1/n)

Enhanced understanding of monotone path emergence

Abstract

We study monotone paths in Erd\H{o}s-R\'enyi random graphs on numbered vertices. Benjamini & Tzalik established a phase transition at $p = \frac{\log n}{n}$ for this model. We refine the critical value to $p = \frac{\log n - \log \log n }{n}$ and identify the critical window of order $\Theta(1/n)$.