The cardinal power of a set A is denoted by IA . Two sets A,, A2 are said to e almost disjoint if IA1 U A2I < lA i l (i = 1, 2) . e call B a transversal of the disjoint non-empty sets A,, (v E M) if Bc U A,, and B intersects each A,. (v E M) in a singleton . D1 An old and well known theorem of W . SIERPINSKI is that an infinite set of po-ver m contains more than m subsets of power m which are pairwise almost disjoint and A . _ARSKI obtained various generalizations and extensions of this in  and  . It is easy to see that Sierpinski's result is equivalent to the following statement : If A, (v E M) are m disjoint sets of power m, then there are snore than m almost disjoint transversals of the A y . In § 3 we prove some new results which are analogous to this rormulation of Sierpinski's theorem . Ve will denote the following statement by _X1 : There are ~2 almost disjoint transversals of t~ r disjoint denumerable sets . In view of recent axiomatic results 4P is independent of the usual axioms of set theory and the generalized continuum hypothesis. In § 4 we show that YE implies a certain unsolved problem of  . In § 5 we consider another question about sets of almost disjoint subsets of a set which was raised by F . S. CATER  .