First order theory on $G(n, c n^{1})$
Abstract
A wellknown result of Shelah and Spencer tells us that the almost sure theory for first order language on the random graph sequence $\left\{G(n, cn^{1})\right\}$ is not complete. This paper proposes and proves what the complete set of completions of the almost sure theory for $\left\{G(n, c n^{1})\right\}$ should be. The almost sure theory $T$ consists of two sentence groups: the first states that all the components are trees or unicyclic components, and the second states that, given any $k \in \mathbb{N}$ and any finite tree $t$, there are at least $k$ components isomorphic to $t$. We define a $k$completion of $T$ to be a first order property $A$, such that if $T + A$ holds for a graph, we can fully describe the first order sentences of quantifier depth $\leq k$ that hold for that graph. We show that a $k$completion $A$ specifies the numbers, up to "cutoff" $k$, of the (finitely many) unicyclic component types of given parameters (that only depend on $k$) that the graph contains. A complete set of $k$completions is then the finite collection of all possible $k$completions.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1802.00059
 Bibcode:
 2018arXiv180200059P
 Keywords:

 Mathematics  Probability
 EPrint:
 This is the version prior to submission. Changes might be incorporated, including an improved title, prior to submission. The updated version will be uploaded soon after