Coherent adequate forcing and preserving CH

We develop a general framework for forcing with coherent adequate sets on H(λ) as side conditions, where λ ≥ ω2 is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of ω2 with finite conditions while preserving CH, solving a problem of Friedman [Forcing with finite conditions, in Set Theory: Centre de Recerca Matematica, Barcelona, 2003–2004, Trends in Mathematics (Birkhauser-Verlag, 2006), pp. 285–295.].