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A well-known result of Shelah and Spencer tells us that the almost suretheory for first order language on the random graph sequence $\left\{G(n,cn^{-1})\right\}$ is not complete. This paper proposes and proves what thecomplete set of completions of the almost sure theory for $\left\{G(n, cn^{-1})\right\}$ should be. The almost sure theory $T$ consists of two sentencegroups: the first states that all the components are trees or unicycliccomponents, and the second states that, given any $k \in \mathbb{N}$ and anyfinite tree $t$, there are at least $k$ components isomorphic to $t$. We definea $k$-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 ofquantifier depth $\leq k$ that hold for that graph. We show that a$k$-completion $A$ specifies the numbers, up to "cutoff" $k$, of the (finitelymany) unicyclic component types of given parameters (that only depend on $k$)that the graph contains. A complete set of $k$-completions is then the finitecollection of all possible $k$-completions.

First order theory on $G(n, c n^{-1})$