We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $\omega$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${ \rm Fin}_k$.
Categories: math.LO math.FA