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We introduce Gowers--Matet forcing with a finite sequence of pairwisenon-isomorphic Ramsey ultrafilters over $\omega$, and with this forcing wesettle the long-standing problem of the spectrum of numbers near-coherenceclasses. We prove that for any finite $n \geq 1$, there is a forcing extensionwith exactly $n$ near-coherence classes of ultrafilters.  For evaluating the new forcing, we prove a strengthening of Gowers's theoremon colourings of ${\rm Fin}_k$.

Local Ramsey Spaces in Matet Forcing Extensions and Finitely Many Near-Coherence Classes