A K0-avoiding dimension group with an order-unit of index two

We prove that there exists a dimension group G whose positive cone is not isomorphic to the dimension monoid DimL of any lattice L. The dimension group G has an order-unit, and can be taken of any cardinality greater than or equal to ℵ2. As to determining the positive cones of dimension groups in the range of the Dim functor, the ℵ2 bound is optimal. This solves negatively the problem, raised by the author in 1998, whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since G has an order-unit of index 2, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to K0, of dimension groups with order-unit of index 2 by unit-regular rings.