A lot of research has been devoted to the study of first order language of graphs (FO). Formulas in this language are constructed using variables $x, y,\dots$ ranging over the vertices of a graph, the usual quantifiers ∀, ∃, the usual logical connectives ¬, ∨, ∧, etc., parentheses and the binary relations =, ∼, where x ∼ y denotes that x and y are adjacent. . In FO one can for instance write “G is triangle-free” as ¬∃x, y, z : (x ∼ y) ∧ (x ∼ z) ∧ (y ∼ z).

In this context, a lot has been studied for 0-1 laws concerning sequences of graphs $\{G_i\}_{i\geq 1}$. For instance, a really strong result due to Shelah and Spencer says that that if $p = n^{−α}$ with $0 ≤ α ≤ 1$ fixed, then the FO-zero-one law holds if and only if α is an irrational number.

Possibly this is obvious for someone coming from logic (I am not from the area), but I would like to know if there are similar models *not* for graphs. In particular, I am interested on knowing if there (or if it has been studied or defined) FOs (or extensions of FOs) over

1.- on subsets of integers (roughly speaking, those are 1-uniform hypergraphs), here it is not clear to me how to define a binary relation. For instance, one would like to encode here that a certain set is $k$-AP free, or a sum-free set.

2.- more generally to $k$-uniform hypergraphs with $k>2$ (definitely we need more than a binary relation).

References would be greatly appreciated