Succinct Definitions in the First Order Theory of Graphs
Oleg Pikhurko, Joel Spencer, Oleg Verbitsky · arXiv (Cornell University) · 2004
We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (resp. D(G)) over all graphs on n vertices. We prove that s(n) and q(n) may be so small that for no general recursive function f we can have f(s(n))\ge n for all n. However, for the function q^*(n)=\max_{i\le n}q(i), which is the least monotone nondecreasing function bounding q(n) from above, we have q^*(n)=(1+o(1))\log^*n, where \log^*n equals the minimum number of iterations of the binary logarithm sufficient to lower n below 1. We show an upper bound q(n)