  For the one dimensional infinite group relaxation, we construct a sequence of
extreme valid functions that are piecewise linear and such that for every
natural number $k\geq 2$, there is a function in the sequence with $k$ slopes.
This settles an open question in this area regarding a universal bound on the
number of slopes for extreme functions. The function which is the pointwise
limit of this sequence is an extreme valid function that is continuous and has
an infinite number of slopes. This provides a new and more refined
counterexample to an old conjecture of Gomory and Johnson stating that all
extreme functions are piecewise linear. These constructions are extended to
obtain functions for the higher dimensional group problems via the
sequential-merge operation of Dey and Richard.
