Intersection Theorems for Systems of Sets (iii)

A system or family (A. : y E N) of sets A y , indexed by the elements of a set N, is called an (a, b)-system if I N I = a and I A,, I = b for y c-N. Expressions such as "(a, < b)-system" are self-explanatory. The system (A,, : y c N) is called a A-system [1] if A,, n A,, = Ap n A Q whenever p, y, p, c c N ; p 7~ y ; p =A a. If we want to indicate the cardinality I NJ of the index set N then we speak of a A(INI)-system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a A(c)-subsystem. In [2], for every choice of cardinals b, c such that b>_2 ; c>_3 ; b+c>==tfi o the least cardinal a = fo(b, c) was determined which has the property that every (a, < b)-system contains a A(c)-subsystem. Let b+ be the least cardinal greater than b. It is easy to see that the following two statements are equivalent : every (a, < b+)-system contains a A(c)-subsystem, every (a, b)-system contains a A(c)-subsystem. In the present note we prove a best possible theorem (Theorem 1) on the size of the largest A-subsystem that can be found in every (m+, m)-system (A, : y E N) which satisfies I Aµ n A v I < n for p, y E N ; p ~ y. Here m > t*~o , and n is a given cardinal, n < m. In proving this theorem the authors have received valuable help from A. Hajnal. We now introduce a condition on systems of sets which is less exacting than that of being a A-system. The system (A Y : y c N) is called a weak A-system (wk 22