ON THE TRANSVERSAL HELLY NUMBERS OF DISJOINT AND OVERLAPPING DISKS BY K. BEZDEK, T. BISZTRICZKY, B. CSIK ´ OS AND A. HEPPES In honour of Helge Tverberg’s seventieth birthday Abstract. A family of disks is said to have the property T (k) if any k mem- bers of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k ≥ 3, the infimum t k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T (k) has a line transversal. We determine exact values of t 3 and t 4 , and give general lower and upper bounds on the sequence t k , showing that t k = O(1/k) as k →∞. 1. Introduction Transversal properties of families of disjoint unit disks have been studied by a number of authors (see references [1]–[15]) with special attention to Helly type problems. For a more recent survey on geometric transversals, we refer to B. Wenger [16]. Consider a finite family F of closed solid circles of diameter 1, called disks, in the plane. We say that the family is t-disjoint if the distance between every pair of centers is larger than t> 0. For t = 1, the disks are disjoint in the original sense. If t> 1 or t< 1, we say that the disks of the family are superdisjoint or overlapping, respectively. A family of disks is said to have property T if the family has a (line ) transversal : a straight line intersecting all members of the family. The family is called a T (k)-family if any k members of the family have a common transversal. If for families of a certain type, property T (k) implies property T but property T (k − 1) does not, we say that families of that type have transversal Helly number k. Danzer [4] proved that for a disjoint family of disks T (5) ⇒ T but T (4) ⇒ T , (see also [1]), thus disjoint families of disks have transversal Helly number 5. Presently, similar problems will be considered for superdisjoint and overlapping families of disks. Our concern is to investigate the relation between t-disjointness and property T (k). It has been found that a lower level of disjointness can be compensated by a higher transversal Helly number; that is, a smaller t by a larger k. We mention that Danzer’s result was extended for disjoint translates of an arbitrary compact convex set in the plane by H. Tverberg [15]. This leads us to the following quite general problem. 1991 Mathematics Subject Classification. 52A35, 52A37, 52A10. Key words and phrases. line transversal, width, disk. 1