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 di- mension grows, the graph becomes similar to an Erdős-Rényi random graph. We pay particular attention to the clique number of such graphs and show that it is very close to that of the corre- sponding Erdo ̋s-Rényi graph when the dimension is larger than log^3 n where n is the number of vertices. The problem is motivated by a statistical problem of testing dependencies.