The Specker-Blatter theorem does not hold for quaternary relations

Let C be a class of relational structures. We denote by fC(n) the number of structures in C over the labeled set {0, . . . , n − 1}. For any C definable in monadic second order logic with unary and binary relation symbols only, E. Specker and C. Blatter showed that for every m ∈ N, the function fC satisfies a linear recurrence relation modulo m, and hence it is ultimately periodic modulo m. The case of ternary relation symbols, and more generally of arity k symbols for k ≥ 3, was left open. In this paper we show that for every m there is a class of structures Cm, which is definable even in first order logic with one quaternary (arity four) relation symbol, such that fCm is not ultimately periodic modulo m. This shows that the Specker-Blatter Theorem does not hold for quaternary relations, leaving only the ternary case open.