Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner

The problem of the partition-numbersJ _?(p, q), considered by Hadwiger and Debrunner for the family ?=C ⁿ of convex bodies, is extended to simplicial complexes and arbitrary families assuming only the validity of Helly’s theorem. We obtain results similar to those of Hadwiger and Debrunner. Further we show the existence of all partition-numbers for the family? = H ⁿC of homothets of a convex body and we get new informations on the partition-numbers for the family of parallel rectangles.