A minimax-measure intersection problem

Abstract

Some years ago, NBS colleague S. Haber communicated the following proble m: To select n subsets of the unit interval, each of mea sw'e 112, so a s to minimize the maximum of the measures of the pairwise inte rsections of these subse ts. The problem is suggested by a paper [1]1 of Gilli s which, settling "an unpubli shed conjec ture of Erdos," proves that for denumerably infinite collections of sets of measure a, the value corresponding to the maximum pairwise-inte rsection meas ure has infimum a 2 • (Collections with higher transfinit e cardinality are treated by Gillis in [2].) Here we provide an explicit solution for collections of finite cardinalities n. Further, and also corresponding to [1], we consider as well the case of p-fold intersections with 2 Sop Son, and provide the corresponding explicit solution. (As noted in [2], the argument of [1] easily extend s to show that aT' is the limiting value for a denumerably infinite collection.) As preliminary, we introduce a second minimization and point out its relationship to our minimax problem, to wit: Select n subsets A 10 A2 , • . , A 1/ of the unit inte rval, eac h of measure a, so that the sum of the measures of their p-fold inte rsections is minimum. If now X = {Slo . . " SI/}' a solution to this minimum proble m, can be chose n so that aU its p-fold intersections have the same measure s, and if M is the maximum of the measures of the p-fold intersections of an arbitrary collection A to A 2, . , " A n with all fL(A J = a, then