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arXiv · 2512.05756

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

Gilles Blanchard·Nicolas Curien·Klara Krause·Alexander Reisach
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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 $\Theta(1/n)$.

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Gilles Blanchard, Nicolas Curien, Klara Krause, Alexander Reisach. 2025-12-05. A phase transition in Erd\H{o}s-Barak random graphs. https://arxiv.org/abs/2512.05756

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