Let ${\cal F}$ denote a family of pairwise disjoint convex
sets in the plane. ${\cal F}$ is said to be in {\em convex
position}, if none of its members is contained in the convex
hull of the union of the others. For any fixed $k\geq 5$, we
give a linear upper bound on $P_k(n)$,
the maximum size of a family ${\cal F}$
with the property that any $k$ members of ${\cal F}$ are in
convex position, but no $n$ are.
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