The Specker–Blatter theorem does not hold for quaternary relations

Abstract

Let C be a class of relational structures. We denote by f C (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 f C 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 C m , which is definable even in first-order logic with one quaternary (arity four) relation symbol, such that f C m 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.