Lower Bounds for Pinning Lines by Balls

A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that every subfamily of size 2d-2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957

Topology of geometric joins

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carath\'eodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is contractible. We are able to prove this when $d$ equals $2$ and $3$, while for larger $d$ we show that the geometric join is contractible provided the number of sets is quadratic in $d$. We also consider a matroid generalization of geometric joins and provide similar bounds in this case

The Erd\H{o}s-Szekeres problem for non-crossing convex sets

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements 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 T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies

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 $f:[0,1] \to \mathbb{R}$. 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