Sh:548

• Shelah, S. (1996). Very weak zero one law for random graphs with order and random binary functions. Random Structures Algorithms, 9(4), 351–358. arXiv: math/9606230 DOI: 10.1002/(SICI)1098-2418(199612)9:4<351::AID-RSA1>3.3.CO;2-D MR: 1605415
• Abstract:
  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 $(x\sim y)$ 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 $L^<$, $\lim_{n\rightarrow\infty}(f_A(n+1)-f_A(n))=0.$

• Version 2011-11-14_12 (8p) published version (9p)

@article{Sh:548,
 author = {Shelah, Saharon},
 title = {{Very weak zero one law for random graphs with order and random binary functions}},
 journal = {Random Structures Algorithms},
 fjournal = {Random Structures \& Algorithms},
 volume = {9},
 number = {4},
 year = {1996},
 pages = {351--358},
 issn = {1042-9832},
 mrnumber = {1605415},
 mrclass = {05C80},
 doi = {10.1002/(SICI)1098-2418(199612)9:4<351::AID-RSA1>3.3.CO;2-D},
 note = {\href{https://arxiv.org/abs/math/9606230}{arXiv: math/9606230}},
 arxiv_number = {math/9606230},
 keyword = {0-1 laws}
}