On Set Systems Having Paradoxical Covering Properties

1. R2-phenomena . Our set theoretic notation will be standard with one exception . Since this paper is largely concerned with powers of ordinals, the symbol ~" will always denote ordinal exponentiation for ordinals ~, 11 . Thus, in particular, if fl--a, then cos is an ordinal <coy+1 . When we use cardinal exponentiation we shall either say so or, if there is no danger of confusion, we write 2 11 0 or Wás (despite the fact that ca),, and RQ otherwise denote the same object) . We shall assume the reader is familiar with the special symbols as defined e .g. in [6] to denote ordinary partition relations, polarized partition relations and square bracket relations . We begin our discussion by recalling a theorem Of MILNER and RADO [13] which asserts that, for any cardinal x--co,