Intersection Theorems for Systems of Sets

A version of Dirichlet's box argument asserts that given a positive integer a and any a2 +1 objects x0 , x1 , . . ., xa 2, there are always a+1 distinct indices v (0 < v < a 2) such that the corresponding ad-1 objects x,, are either all equal to each other or mutually different from each other . This proposition can be restated as follows . Let N be an index set of more than a 2 elements and let, for each element v of N, X v be a one-element set . Then there is a subset N' of N having more than a elements, such that all intersections X, X, corresponding to distinct elements μ, v of N' have the same value . In this note we investigate extensions of this principle to cases when the sets X„ are of any prescribed cardinal b . Both a and bare given cardinals, finite or infinite . In the case of finite a and b we obtain estimates for the number which corresponds to a2 in Dirichlet's case, and we show that when at least one of a and b is infinite then av+ 1 is the best possible value of that number . We introduce some definitions ;' . A system E1 : Y v (v e N) of sets Y,, where v ranges over the index set N, is said to contain the system 10 : N1,. (μ c 31) if, for every μo of 111, the set X., occurs in 'r''1 at least as often as in Eo , i .e . if