Pro Set Theory

which means that the following is true : If ISI= a and [S]r= {X S :I I = r) = K oUK,, then there are TcS and i< 2 such that ITI bb and [T] rC K .-The ymbol (2)w s first used in a p per 2 Z by Rado and my lf (1952) ut a number of papers d aling with generaliz tions of Ram y's theorem appeared before then . The first these was nl. a paper by W . Sierpinski who prove that 2 0+* (~ I , k 1 ) 2 . References to th earlier literatu will be found in [b d [21 Finally n [21 he truth value of (2) is determined for arbitrary cardinals a, b o , bl provided th g neral zed cont uum hypothesis is ssured and ha a is not larger than the fir t inaccessible cardin l . Here I want to state just on p oblem . Withou assu ing the co tinuum hypothes , is it p sibl t plit the pairs of r al numbers into three classes so that in any uncountable subset of the reals there is a pa r in e ch of the three c asses? Wi the continuum hypothes this and mu more is true (see [2]) .