Exponential tails for factors and the chromatic number of random graphs

Abstract

The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more generally, the probability that the largest $K_r$-matching covers less than $n-s$ vertices of $G(n,p)$. For every fixed $r\ge3$, throughout the range $$n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},\qquad n-s\in r\mathbb Z,\qquad s=o(n),$$ we prove $$\mathbb P\bigl(\phi_r^s(G(n,p))=0\bigr)=\exp\left(-\Theta_r\!\left((s+1)\frac{\mu_r(n,p)}n\right)\right),$$ where $\phi_r^s(G)$ is the number of $K_r$-matchings covering exactly $n-s$ vertices and $\mu_r(n,p):=\binom nrp^{\binom r2}$. The lower bound is given by $s+1$ vertices which lie in no copy of $K_r$. For the upper bound we develop an iterable one-root version of the Johansson--Kahn--Vu method. As a structural consequence, we show that the remainder of $G(n,p)$ outside every maximal $K_r$-matching has an almost-perfect $K_{r-1}$-matching throughout the sparse clique window. Independently, we prove a central limit theorem for the maximum $K_r$-matching number. Combining these inputs and a structural theorem for $r=2$ from our earlier work, we prove a central limit theorem for the chromatic number of very dense random graphs: for every $r\ge2$ and $n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},$ $$\frac{\chi(G(n,1-p))-\mathbb E\chi(G(n,1-p))}{\sqrt{\mu_{r+1}(n,p)}/r}\xrightarrow{\mathrm d}\mathcal N(0,1),\qquad\operatorname{Var}\bigl(\chi(G(n,1-p))\bigr) \sim\frac{\mu_{r+1}(n,p)}{r^2}.$$ This settles the Surya--Warnke conjecture throughout the interior of every clique window with $r\ge2$, strengthening its concentration prediction to a Gaussian limit with asymptotically exact variance.