Geodesic
On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture
The electronic journal of combinatorics, Tome 25 (2018) no. 3
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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 $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues 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 analogues of linear thrackles and conjecture their continuous generalizations.
DOI : 10.37236/6516
Classification : 52A35, 05C10, 68R10, 52A10
Mots-clés : Tverberg's theorem, Reay's conjecture, thrackles 
@article{10_37236_6516,
     author = {Megumi Asada and Ryan Chen and Florian Frick and Frederick Huang and Maxwell Polevy and David Stoner and Ling Hei Tsang and Zoe Wellner},
     title = {On {Reay's} relaxed {Tverberg} conjecture and generalizations of {Conway's} thrackle conjecture},
     journal = {The electronic journal of combinatorics},
     year = {2018},
     volume = {25},
     number = {3},
     doi = {10.37236/6516},
     zbl = {1397.52004},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/6516/}
}
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%A Zoe Wellner
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Megumi Asada; Ryan Chen; Florian Frick; Frederick Huang; Maxwell Polevy; David Stoner; Ling Hei Tsang; Zoe Wellner. On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture. The electronic journal of combinatorics, Tome 25 (2018) no. 3. doi: 10.37236/6516

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