In a previous paper with M. Džamonja, class forcings were given which fixed the complexity (a universality covering number) for certain types of structures of size λ together with the value of 2λ for every regular λ. As part of a programme for examining when such global results can be true in an inner model, we build generics for these class forcings.
"Internal consistency for embedding complexity." J. Symbolic Logic 73 (3) 831 - 844, September 2008. https://doi.org/10.2178/jsl/1230396750