Sunflowers of Convex Open Sets

A sunflower is a collection of sets $\{U_1,\ldots, U_n\}$ such that the pairwise intersection $U_i\cap U_j$ is the same for all choices of distinct $i$ and $j$. We study sunflowers of convex open sets in $\mathbb R^d$, and provide a Helly-type theorem describing a certain "rigidity" that they possess. In particular we show that if $\{U_1,\ldots, U_{d+1}\}$ is a sunflower in $\mathbb R^d$, then any hyperplane that intersects all $U_i$ must also intersect $\bigcap_{i=1}^{d+1} U_i$. We use our results to describe a combinatorial code $\mathcal C_n$ for all $n\ge 2$ which is on the one hand minimally non-convex, and on the other hand has no local obstructions. Along the way we further develop the theory of morphisms of codes, and establish results on the covering relation in the poset $\mathbf P_{\mathbf{Code}}$

Distances between realizations of order types

Any $n$-tuple of points in the plane can be moved to any other $n$-tuple by a continuous motion with at most $\binom{n}{3}$ intermediate changes of the order type. Even for tuples with the same order type, the cubic bound is sharp: there exist pairs of $n$-tuples of the same order type requiring $c\binom{n}{3}$ intermediate changes.Comment: 9 pages, 1 figur

Quantitative upper bounds on the Gromov-Hausdorff distance between spheres

The Gromov-Hausdorff distance between two metric spaces measures how far the spaces are from being isometric. It has played an important and longstanding role in geometry and shape comparison. More recently, it has been discovered that the Gromov-Hausdorff distance between unit spheres equipped with the geodesic metric has important connections to Borsuk-Ulam theorems and Vietoris-Rips complexes. We develop a discrete framework for obtaining upper bounds on the Gromov-Hausdorff distance between spheres, and provide the first quantitative bounds that apply to spheres of all possible pairs of dimensions. As a special case, we determine the exact Gromov-Hausdorff distance between the circle and any even-dimensional sphere, and determine the asymptotic behavior of the distance from the 2-sphere to the $k$-sphere up to constants.Comment: 13 pages, 3 figure

Embedding dimension gaps in sparse codes

We study the open and closed embedding dimensions of a convex 3-sparse code $\mathcal{FP}$, which records the intersection pattern of lines in the Fano plane. We show that the closed embedding dimension of $\mathcal{FP}$ is three, and the open embedding dimension is between four and six, providing the first example of a 3-sparse code with closed embedding dimension three and differing open and closed embedding dimensions. We also investigate codes whose canonical form is quadratic, i.e. ``degree two" codes. We show that such codes are realizable by axis-parallel boxes, generalizing a recent result of Zhou on inductively pierced codes. We pose several open questions regarding sparse and low-degree codes. In particular, we conjecture that the open embedding dimension of certain 3-sparse codes derived from Steiner triple systems grows to infinity.Comment: 16 pages, 8 figure

Enumeration of interval graphs and dd-representable complexes

For each fixed $d\ge 1$, we obtain asymptotic estimates for the number of $d$-representable simplicial complexes on $n$ vertices as a function of $n$. The case $d=1$ corresponds to counting interval graphs, and we obtain new results in this well-studied case as well. Our results imply that the $d$-representable complexes comprise a vanishingly small fraction of $d$-collapsible complexes.Comment: 12 pages, 2 figure

Colorful words and dd-Tverberg complexes

We give a complete combinatorial characterization of weakly $d$-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in $\mathbb R^d$. This strengthens the concept of $d$-representable complexes, which describe intersection combinatorics that arise in at least one point set. Our characterization allows us to construct for every fixed $d$ a graph that is not weakly $d'$-Tverberg for any ${d' \le d}$, answering a question of De Loera, Hogan, Oliveros, and Yang.Mathematics Subject Classifications: 52A35, 52C45Keywords: Tverberg's theorem, word representable, $d$-representable, nerve, general position, strong general position, fully independen