Upper Tails for Cliques

Details Upper Tails for Cliques Upper Tails for Cliques

2011-11-29

arxiv.org/abs/1111.6687v1

Mathematics

Probability

25 pages

Scientific paper

With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual
(Erd\H{o}s-R\'enyi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$,
we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k}
\min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$ This is tight up to the
value of the constant in the exponent.

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