Some Remarks on Set Theory, Ix . Combinatorial Problems in Measure Theory and Set Theory

Now, in analogy to Ramsay's theorem, one might consider the following problem. Suppose that, for some u > 0, there is associated with each k-tuple X = {x l , • • • , xk } of elements of an infinite set S a measurable set F(X) of [0, 1] such that m(F(X)) > u . Does there always exist an infinite subset S' of S such that the sets F(X) corresponding to the k-tuples X of S' have a nonempty intersection? We study this and related questions . In the course of our investigation we are led to a surprising number of unsolved problems .