On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

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.

[1]  Pavle V. M. Blagojevi'c,et al.  Optimal bounds for the colored Tverberg problem , 2009, 0910.4987.

[2]  Jean-Paul Doignon,et al.  Radon partitions and a new notion of independence in affine and projective spaces , 1977 .

[3]  Florian Frick,et al.  Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints , 2015, Journal of the European Mathematical Society.

[4]  Florian Frick,et al.  Counterexamples to the topological Tverberg conjecture , 2015 .

[5]  Micha A. Perles,et al.  Some variations on Tverberg’s theorem , 2016 .

[6]  H. Tverberg.  A Generalization of Radon's Theorem , 1966 .

[7]  M. Perles,et al.  A generalization of Tverberg's Theorem , 2007, 0710.4668.

[8]  János Pach,et al.  A computational approach to Conway's thrackle conjecture , 2011, Comput. Geom..

[9]  J. R. Reay.  Several generalizations of Tverberg’s theorem , 1979 .

[10]  László Lovász,et al.  On Conway's Thrackle Conjecture , 1995, SCG '95.

[11]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems , 2015, ArXiv.

[12]  John R. Reay.  Twelve general position points always form three intersecting tetrahedra , 1979, Discret. Math..

[13]  János Pach,et al.  A Computational Approach to Conway's Thrackle Conjecture , 2010, Graph Drawing.

[14]  Micha A. Perles,et al.  Strong General Position , 2014 .

[15]  Imre Bárány,et al.  Tverberg’s Theorem at 50: Extensions and Counterexamples , 2016 .

[16]  Micha A. Perles,et al.  Tverberg Partitions of Points on the Moment Curve , 2017, Discret. Comput. Geom..

[17]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[18]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[19]  I. Bárány,et al.  A Colored Version of Tverberg's Theorem , 1992 .