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

by Shelah. [Sh:548]

Random Structures \& Algorithms, 1996

Let G_<(n,p) denote the usual random graph G(n,p)
on
a
totally ordered set of n vertices. We will fix
p=
frac
{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_A(n) denote
the
probability that the random G_<(n,p) has property
A
.
It
is
known Compton, Henson and Shelah [CHSh:245] that
there
are
A
for
which f_A(n) does not converge. Here we show what
is
called
a
very weak zero-one law (from [Sh 463]):
THEOREM: For every A in language L^<, lim_{n->
infty}(f_A(n+1)-f_A(n))=0.

Back to the list of publications