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Regular systems of paths and families of convex sets in convex position
In this paper we show that every sufficiently large family of convex bodies
in the plane has a large subfamily in convex position provided that the number
of common tangents of each pair of bodies is bounded and every subfamily of
size five is in convex position. (If each pair of bodies have at most two
common tangents it is enough to assume that every triple is in convex position,
and likewise, if each pair of bodies have at most four common tangents it is
enough to assume that every quadruple is in convex position.) This confirms a
conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes
Toth. Our results on families of convex bodies are consequences of more general
Ramsey-type results about the crossing patterns of systems of graphs of
continuous functions . On our way towards proving the
Pach-Toth conjecture we obtain a combinatorial characterization of such systems
of graphs in which all subsystems of equal size induce equivalent crossing
patterns. These highly organized structures are what we call regular systems of
paths and they are natural generalizations of the notions of cups and caps from
the famous theorem of Erdos and Szekeres. The characterization of regular
systems is combinatorial and introduces some auxiliary structures which may be
of independent interest
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