Speaker: Andreas Holmsen

Title: On Katchalski's fractional transversal problem

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 talk 
we will discuss some basic properties of T(k)-families: Finite familis of 
convex sets in the plane in which every subfamily of size at most k admits 
a common transversal. In 1980 Kathalski and Liu showed that for every 
k/geq 3 there is a fraction a(k) such that any T(k)-family F admits a 
partial transversal of size a(k)|F|, and furthermore that a(k) tends to 1 
as k tends to infinity. Prior to this, Katchalski made a conjecture about 
the initial value: a(3) = 2/3. In this talk we will review some related 
results from geometric transversal theory: Eckhoff's theorems for T(3) and 
T(4) families and the recent colorful Hadwiger theorem due to Arocha, 
Bracho adn Montejano. We also give a simple argument providing the 
sharpest known lower bound for a(k), as well as a constructions which show 
that a(3) \leq 0.53989... and a(4) \leq 0.76502.