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First Order Definability of Trees and Sparse Random Graphs

Tom Bohman·Alan Frieze·Tomasz Luczak·Oleg Pikhurko·Clifford Smyth·Joel Spencer·Oleg Verbitsky
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Abstract

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 first order descriptive complexity of G. We will 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.

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Tom Bohman, Alan Frieze, Tomasz Luczak, Oleg Pikhurko, Clifford Smyth, Joel Spencer, Oleg Verbitsky. 2005-06-15. First Order Definability of Trees and Sparse Random Graphs. https://arxiv.org/abs/math/0506288

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