Large finite structures with few Lκ-types

Far each κ⩾3, we show that there is no recursive bound for the size of the smallest finite model of an L^κ-theory in terms of its κ-size. Here L^κ denotes the κ-variable fragment of first-order logic. An L^κ-theory is a maximal consistent set of L ^κ-sentences, and the κ-size of an L^κ -theory is the number of L^κ-types realized in its models. Our result answers a question of Dawar (1993). As a corollary, we obtain that for κ⩾3 the so-called L^κ-invariants, which characterize structures up to equivalence in L^κ, cannot be recursively inverted