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We show an equivalence between a conjecture of Bisztriczky and Fejes Tóth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and Tóth on the Erdős–Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős–Szekeres theorem for non‐crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős–Szekeres theorem for disjoint (and non‐crossing) convex bodies, as well as a generalization of the partitioned Erdős–Szekeres theorem of Pór and Valtr to families of non‐crossing convex bodies.

THE ERDŐS–SZEKERES PROBLEM FOR NON‐CROSSING CONVEX SETS