Very weak zero one law for random graphs with order and random binary functions

Author: Saharon Shelah

ABSTRACT

Let G(<,n,p) denote the usual random graph G(n,p) on a totally ordered set of n vertices. (We naturally think of the vertex set as 1,...,n with the usual <). We will fix p=1/2 for definiteness. Let L(<) denote the first order language with predicates equality (x=y), adjacency (x~y) and less than (x < y). For any sentence A in L(<) let f(n)=f(A,n) denote the probability that the random G(<,n,p) has property A. It is known that there are A for which f(n) does not converge. Here we show what is called "a very weak zero-one law"

Theorem: For every A in language L(<) lim [f(A,n+1)-f(A,n)] = 0.

Note, as an extreme example, that this implies the nonexistence of a sentence A holding with probability 1-o(1) when n is even and with probability o(1) when n is odd.

In section 2 we give the proof, based on a circuit complexity result. In Section 3 we prove that result, which is very close to the now classic theorem that parity cannot given by an AC^0 circuit. In Section 4 we give a very weak zero-one law for random two-place functions. The proof is very similar, the random function theorem being perhaps of more interest to logicians, the random graph theorem to discrete mathematicians.

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