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 α ∣ F ∣ for some constant α ( k ) which is independent of F . Here it will be shown that (2/( k ( k −1))) 1/( k −2) ≤ α ( k )≤(( k −2)/( k −1)), which are the best bounds to date.
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