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arXiv · 1005.4471

Upper tails for triangles

Bobby DeMarco·Jeff Kahn
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Abstract

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) $$\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].$$ This is tight up to the value of $C_{\eta}$.

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Bobby DeMarco, Jeff Kahn. 2010-05-25. Upper tails for triangles. https://doi.org/10.1002/rsa.20382

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