On the Structure of Set-mappings

1 . Introduction. Let S be a set and f(x) a function which makes correspond to every x E S a subset fix) of S so that x(f (x) . Such a function f(x) we shall call a set-mapping defined on S. A subset S' 9 S is called free (or independent) with respect to the set-mapping f (x), if for every x E S' and Y E S', x ~f(y) and y ~ ftx) . Let S = m o and n < m. Assume that f(x) < n for every x E S. RUZIEWICZ raised the problem if there always exists a free set of power m . Assuming the generalized hypothesis of the continuum the answer to the problem of RUZIEWICZ is positive .' In our present paper we are going to define more general set-mappings and raise analogous questions to those of RUZIEWICZ . Let there be given the set S and a set of its subsets L Assume that the function f (X) makes correspond to every set X of 1 a subset f (X) of S so that the intersection of f(X) and X is empty . f(X) will be defined as a set-mapping of S of type L This is clearly a generalization of the original concept of the set-mapping . (There 1 consisted of all the subsets of S having one element .) The subset S' 9;S is called free (with respect to this set-mapping, or briefly a free set of the set-mapping) if for every X 9 S' (X E I) the intersection of f(X) and S' is empty . Our aim is the investigation of those set-mappings where I consists either of all subsets of S of a given cardinal t or of all subsets of less than a given cardinal t. In these cases we shall briefly say that the set-mapping is of type t or of type < t, respectively . If f(X) < n for all X of I we shall say that the set-mapping is of order n. Our problems will be of the following kind : Let S = m, further let f(X) be a set-mapping of S of order n and type t. Does there then always exist an independent set of power p?