Easton's theorem for Ramsey and strongly Ramsey cardinals

We show that, assuming GCH, if is a Ramsey or a strongly Ramsey cardinal and F is a class function on the regular cardinals having a closure point at and obeying the constraints of Easton's theorem, namely, F ( ) F ( ) for and < cf(F ( )), then there is a conality preserving forcing extension in which remains Ramsey or strongly Ramsey respectively and 2 = F ( ) for every regular cardinal .

[1]  Joel David Hamkins,et al.  What is the theory ZFC without power set? , 2011, Math. Log. Q..

[2]  A. Kanamori.  The higher infinite : large cardinals in set theory from their beginnings , 2005 .

[3]  Joel David Hamkins,et al.  What is the theory ZFC without power set? , 2011, Math. Log. Q..

[4]  Fred Galvin,et al.  On the Singular Cardinals Problem , 1981 .

[5]  Andras Hajnal,et al.  Some remarks concerning our paper „On the structure of set-mappings” —Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal , 1962 .

[6]  Andrzej Zarach.  Unions of Zf--Models Which are Themselves Zf--Models , 1982 .

[7]  Gregory H. Moore.  Early history of the Generalized Continuum Hypothesis: 1878 - 1938 , 2011, Bull. Symb. Log..

[8]  Telis K. Menas,et al.  Consistency results concerning supercompactness , 1976 .

[9]  A. Kanamori.  The Higher Infinite , 1994 .

[10]  J. Cummings.  Iterated Forcing and Elementary Embeddings , 2010 .

[11]  William B. Easton,et al.  Powers of regular cardinals , 1970 .

[12]  Philip D. Welch,et al.  Ramsey-like cardinals II , 2011, J. Symb. Log..

[13]  Jean-Pierre Levinski.  Filters and Large Cardinals , 1995, Ann. Pure Appl. Log..

[14]  Kenneth Kunen,et al.  Some applications of iterated ultrapowers in set theory , 1970 .

[15]  R. Jensen,et al.  The Core Model , 1982, An Introduction to Geographical and Urban Economics.

[16]  Victoria Gitman.  Ramsey-like cardinals , 2011, J. Symb. Log..

[17]  Moti Gitik.  On Measurable Cardinals Violating the Continuum Hypothesis , 1993, Ann. Pure Appl. Log..

[18]  Arthur W. Apter.  On some questions concerning strong compactness , 2012, Arch. Math. Log..

[19]  Brent Cody.  Some Results on Large Cardinals and the Continuum Function , 2012, 1209.1136.

[20]  Sy-David Friedman,et al.  Easton's theorem and large cardinals , 2008, Ann. Pure Appl. Log..

[21]  William J. Mitchell.  Ramsey Cardinals and Constructibility , 1979, J. Symb. Log..

[22]  V. Gitman.  Applications of the proper forcing axiom to models of Peano arithmetic , 2007 .