The neighborhood complex of a random graph

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For a graph $G$, the neighborhood complex $N[G]$ is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovász that if $N[G]$ is $k$-connected, then the chromatic number of $\chi(G) \ge k+3$. We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between $1/2$ and $2/3$ of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, $O(\log d)$, compared to the expected dimension $d$ of the complex itself.