Small graph classes and bounded expansion

A class of simple undirected graphs is small if it contains at most n ! α n labeled graphs with n vertices, for some constant α. We prove that for any constants c , ε > 0 , the class of graphs with expansion bounded by the function f ( r ) = c r 1 / 3 − ε is small. Also, we show that the class of graphs with expansion bounded by 6 ⋅ 3 r log ( r + e ) is not small.