AbstractFor any k≥3, we prove the following results on theories in the k-variable fragment Lk of first-order logic:(1) The so-called Lk-invariant is not recursively invertible.(2) There is no recursive bound on the size of the minimal model of a complete Lk-theory in terms of its k-size, that is, the number of its Lk-types.(3) Assume that NP⊈P/poly. Then there is no P-computable canonization function for Lk-equivalence.Our results answer questions of Dawar [3, 4] and Dawar, Lindell, and Weinstein [5].