A phase transition in Erdős-Barak random graphs

preprint2026arXivOpen access

We study monotone paths in Erdős-Rényi 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 $Θ(1/n)$ .

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A phase transition in Erdős-Barak random graphs

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