Modal counterparts of Medvedev logic of finite problems are not finitely axiomatizable

We consider modal logics whose intermediate fragments lie between the logic of infinite problems 20] and the Medvedev logic of finite problems 15]. There is continuum of such logics 19]. We prove that none of them is finitely axiomatizable. The proof is based on methods from 12] and makes use of some graph-theoretic constructions (operations on coverings, and colourings).