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{\bf Jan Kyn\v{c}l and Martin Tancer}
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{\bf The Maximum Piercing Number for some Classes of Convex Sets with the $(4,3)$-property}
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A finite collection ${\cal C}$ of closed convex sets in ${\Bbb R}^d$
is said to have a {\em $(p,q)$-property}\/ if among any $p$ members of
${\cal C}$ some $q$ have a non-empty intersection, and $|{\cal C}|
\ge p$. A {\em piercing number}\/ of ${\cal C}$ is defined as the
minimal number $k$ such that there exists a $k$-element set which
intersects every member of ${\cal C}$. We focus on the simplest
non-trivial case in ${\Bbb R}^2$, i.e., $p=4$ and $q=3$. It is known
that the maximum possible piercing number of a finite collection of
closed convex sets in the plane with $(4,3)$-property is at least $3$
and at most $13$. We consider the following three special types of
collections of closed convex sets: segments in ${\Bbb R}^d$, unit
discs in the plane and positively homothetic triangles in the plane,
in each case only those satisfying $(4,3)$-property.  We prove that
the maximum possible piercing number is $2$ for the collections of
segments and $3$ for the collections of the other two types.

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