Large Finite Structures with Few Lk-Types

For each k 3, we show that there is no recursive bound for the size of the smallest finite model of an Lk-theory in terms of its k-size. Here Lk denotes the k-variable fragment of first-order logic. An Lk-theory is a maximal consistent set of Lk-sentences, and the k-size of an Lk-theory is the number of Lk-types realized in its models. Our result answers a question of Dawar [3, 2]. As a corollary, we obtain that for k 3 the so-calledLkinvariants, which characterize structures up to equivalence in Lk, cannot be recursively inverted.