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In this paper we show that every sufficiently large family of convex bodiesin the plane has a large subfamily in convex position provided that the numberof common tangents of each pair of bodies is bounded and every subfamily ofsize five is in convex position. (If each pair of bodies have at most twocommon 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 isenough to assume that every quadruple is in convex position.) This confirms aconjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and FejesToth. Our results on families of convex bodies are consequences of more generalRamsey-type results about the crossing patterns of systems of graphs ofcontinuous functions $f:[0,1] \to \mathbb{R}$. On our way towards proving thePach-Toth conjecture we obtain a combinatorial characterization of such systemsof graphs in which all subsystems of equal size induce equivalent crossingpatterns. These highly organized structures are what we call regular systems ofpaths and they are natural generalizations of the notions of cups and caps fromthe famous theorem of Erdos and Szekeres. The characterization of regularsystems is combinatorial and introduces some auxiliary structures which may beof independent interest.

Regular systems of paths and families of convex sets in convex position