Random Graphs, Retractions and Clique Graphs

We found d 1 = d 1 ( n , p ) and d 2 = d 2 ( n , p ) such that almost every (random) graph G ∈ G ( n , p ) has retractions to d-dimensional octahedra O d for every integer d satisfying d 1 < d < d 2 . This result has several important consequences: (the clique complex of) the random graph has several non trivial homology/homotopy groups, the random graph is not contractible, the random graph is not homotopy equivalent to its clique graph and the random graph is clique divergent.