关于Reay放松的Tverberg猜想和Conway Thrackle猜想的推广
On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture
摘要：
Reay的放松Tverberg猜想和Conway的Thrackle猜想是关于两两相交几何的公开问题。雷伊要求的最低数量的点在欧几里德d-空间，保证任何这样的点集承认一个分区成r部分，任何k的凸包相交。在这里我们给出了这个数的新的和改进的下界，Reay证明了它与k无关。我们证明了一个彩色版本的雷伊猜想k足够大，但k独立的维度d。要求凸包成对相交严重限制了组合数学。这是一个高维的类似康威的thrackle猜想或其线性特殊情况。因此，我们研究凸几何和高维类似物的thrackle猜想旁边雷伊的问题和猜想（并证明在两个特殊情况下），在平面上的凸集的数量是有界的顶点总数，他们涉及每当存在一个横向集为他们的两两相交。因此，我们孤立的几何性质，导致在thrackle猜想的界限。我们还建立了严格的界限的高维类似物的线性thrackles和猜想他们的连续推广方面的数量。
英文摘要：
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.
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