First-Order Definability of Trees and Sparse Random Graphs

Let D(G) be the smallest quantifier depth of a first order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first order logic. We show that almost surely D(G) = Θ( ln n ln ln n ), where G is a random tree of order n or the giant component of a random graph G(n, c n ) with constant c > 1. These results rely on computing the maximum of D(T ) for a tree T of order n and maximum degree l, so we study this problem as well.