On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture

TL;DR

This paper advances understanding of geometric intersection problems by providing new bounds for Reay's relaxed Tverberg conjecture and exploring higher-dimensional analogs of Conway's thrackle conjecture, including special case proofs and tight bounds.

Contribution

It offers improved lower bounds for Reay's conjecture, proves a colored version for large k, and establishes tight bounds for higher-dimensional thrackle analogs.

Findings

New lower bounds for Reay's conjecture.

Proof of a colored version of Reay's conjecture for large k.

Tight bounds for facets of higher-dimensional thrackle analogs.

Abstract

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean d-space that guarantees any such point set admits a partition into r parts, any k of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of k. We prove a colored version of Reay's conjecture for k sufficiently large, but nevertheless k independent of dimension d. Requiring convex hulls to intersect pairwise severely restricts combinatorics. This is a higher-dimensional analog of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogs of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogs of linear thrackles and conjecture their continuous generalizations.