NEW RESULTS FOR T (k)-FAMILIES IN THE PLANE

Abstract

A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F. In this paper we study some basic properties of T (k)-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T (k)-family admits a partial transversal of size alpha vertical bar F vertical bar for some constant alpha(k) which is independent of F. Here it will be shown that (2/(k(k - 1)))(1/(k-2)) <= alpha(k) <= ((k - 2)/k - 1)), which are the best bounds to date.