High-dimensional random geometric graphs and their clique number
We study the behavior of random geometric graphs in high dimensions. We show that as the dimension grows, the graph becomes similar to an Erd ̋os-R ́enyi random graph. We pay particular attention to the clique number of such graphs and show that it is very close to that of the corresponding Erd ̋os-R ́enyi graph when the dimension is larger than log3 n where n is the number of vertices. The problem is motivated by a statistical problem of testing dependencies.