The Penultimate Rate of Growth for Graph Properties

Given a property P of graphs, write Pn for the set of graphs with vertex set [n] having property P. We call |Pn| the speed of P. Recent research has shown that the speed of a monotone or hereditary property P can be a constant, polynomial, or exponential function of n, and the structure of the graphs in P can then be well described. Similarly, |Pn| can be of the form n(1−1/k+o(1))n or 2(1−1/k+o(1))n 2/2 for some k > 1 and the properties can be described and have well behaved speeds. In this paper, we discuss the behavior of properties with speeds between these latter bounds, i.e. between n(1+o(1))n and 2(1/2+o(1))n /2.