Collapsibility of random clique complexes

We prove a sufficient condition for finite clique complex to collapse k-dimensional complex, and use this exhibit thresholds (k+1)-collapsibility in sparse random complex. In particular, if every strongly connected, pure (k+1)-dimensional subcomplex of X has vertex degree at most 2k+1, then is (k+1)-collapsible. the model X(n,p) complexes an Erdős–Rényi graph G(n,p), we show that any fixed k≥0, p=n−α α>1/(k+1), X=distX(n,p) (k+1)-collapsible with high probability.2