This is a spiritual successor to a question that Peter Shor answered here:


[Generalized Euclidean TSP](https://mathoverflow.net/questions/57017/generalized-euclidean-tsp)


Are there any results known on the asymptotic behavior of clique sizes in a unit disk graph with uniformly sampled points? That is, suppose I sample $N$ points uniformly in a square (or disk, or whatever figure is easiest to deal with) of size $L \times L$, and I draw an edge between two points if the distance between them is less than $1$. As $N$ becomes large, is anything known about the distribution of clique sizes in this graph?



Clique sizes in a unit disk graph
Yes, a lot is known about this. See Mathew Penrose's book, "Random Geometric Graphs." He discusses subgraph counts for random geometric graphs on distributions with bounded, measurable, density functions --- this includes uniform distributions on any convex body as a special case. He also discusses maximum clique size.


See also his recent paper with Yukich, [Limit theory for point processes in manifolds](https://arxiv.org/abs/1104.0914).


