Happy families and completely Ramsey sets

Summary We use games of Kastanas to obtain a new characterization of the classC of all sets that are completely Ramsey with respect to a given happy family . We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC. We also study some cardinals that can be associated with the ideal of nowhere -Ramsey sets.Part of this research was done while the author was visiting I.V.I.C. in Caracas in September 1989. The author would like to thank Carlos Di Prisco for his hospitality.