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.