# Blatter-Specker Theorem for ternary relations

Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .

**Question**Does the Blatter-Specker Theorem hold for ternary relations.

Our exposition follows closely [BS84].

## Counting labeled structures modulo

Let be a class of finite structures for one binary relation symbol . We define for

#### Examples:

- \item If consists of all -structures, . \item If consists of bijections, \item If is the class of all (undirected, simple) graphs, . \item If is the class of all equivalence relations, then , the {\em Bell Numbers}. \item If is the class of all equivalence relations with two classes only, of the same size, . Clearly, . \item If is the class of all trees, , {\em Caley}.

We observe the following:

And for each the functions, , , are ultimately periodic .

However, iff , hence is not periodic .

## Monadic second-order logic definable classes

The first four examples (all relations, all bijections, all graphs, all equivalence relations) are definable in First Order Logic . The trees are definable in Monadic Second Order Logic ..

is definable in Second Order Logic , but it is not -definable. If we expand to have the bijection between the classes we get structures with two binary relations. The class is now -definable. Let us denote the corresponding counting function . We have for large enough.

## Periodicity and linear recurrence relations

The periodicity of is usually established by exhibiting a *linear recurrence relation*:

There exists and integers such that for all

#### Examples.

- \item In the case of we have \item In the case of we have for all In this case we say that

*trivializes*.

## The Blatter-Specker Theorem

**Theorem (BS84)**Let be a binary vocabulary, i.e. all relation symbols are at most binary. If is a class of finite -structures which is -definable, then for all is ultimately periodic .

Moreover, there exists and integers such that for all i.e we have a linear recurrence relation.

In [F03] Fischer showed that the Specker-Blatter Theorem does not hold for quaternary relations.

The case of ternary relations remains open.

See also [FM06] for further developments on the topic.

## Bibliography

[BS84]* C. Blatter and E. Specker, Recurrence relations for the number of labeled structures on a finite set, Logic and Machines: Decision Problems and Complexity, E. Börger, G. Hasenjaeger and D. Rödding, eds, LNCS 171 (1984) pp. 43-61.

[F03] E. Fischer, The Specker-Blatter theorem does not hold for quaternary relations, Journal of Combinatorial Theory Series A 103(2003), 121-136.

[FM06] E. Fischer and J. A. Makowsky, The Specker-Blatter Theorem revisited: Generating functions for definable classes of stuctures. In Computing and Combinatorics (COCOON 2003) Proc., LNCS vol. 2697 (2003), 90-101.

[S88] E. Specker, Application of Logic and Combinatorics to Enumeration Problems, Trends in Theoretical Computer Science, E. Börger ed., Computer Science Press, 1988, pp. 141-169. Reprinted in: Ernst Specker, Selecta, Birkhäuser 1990, pp. 324-350.

* indicates original appearance(s) of problem.