To begin to explain the lines on the graph we recall that Sawyer (1979) showed the size of the opinion at the origin O divided by its mean converges to an exponential distribution with mean 1. Picking the opinion containing O gives k opportunities for a cluster of size k. Removing this size bias from Sawyer's result leads easily to a result for N(i,j), the number of types with sizes in [i,j). Here and in what follows

and we suppose
for some c > 0.

Keeping to the numbering in our papers we now state:

Theorem 3. For any 0 < u < v

The curve of triangles shows the predictions for this result which fit the data closely for cells 8 and above.