Almost sure theories

If

is a model with universe $\text{U}\text{and}\text{Q ⊆}{}^{\mathrm{q}}\text{U}$ where q is a fixed positive integer, we put

〈Q〉 for the expansion of

with the new relation Q. We study sets of relations defined by $\text{S(σ) = {Q⊆}{}^{\mathrm{q}}\text{U:}\textcolor{red}{}\text{〈Q〉⊨σ}}$ where σ is a first-order sentence with equality of the appropriate type and $\text{|U|⩽ℵ}{}_0$. For some simple countable structures

, we show that S(σ) is almost all of

2 or almost none of it, for certain topologies and measures. We have analogous results for the cardinality of S(σ) for some finite structures

with large enough U.

Some of the structures we examine, in both the countable and finite case, are sets with a successor relation and cyclic groups.