Tverberg-type theorems for intersecting by rays

Intersecting diametral balls induced by a geometric graph

For a graph whose vertex set is a finite set of points in the Euclidean $d$-space consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that ($i$) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; ($ii$) for any $ n $ red and $ n $ blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Also, we prove that ($iii$) for any even set of points in the Euclidean $ d $-s…

Optimal colored Tverberg theorems for prime powers

The type A colored Tverberg theorem of Blagojević, Matschke, and Ziegler provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices is a prime number. We extend this result to an optimal, type A colored Tverberg theorem for multisets of colored points, which is valid for each prime power $r=p^k$. One of the principal new ideas is to replace the ambient simplex $Δ^N$, used in the original Tverberg theorem, by an "abridged simplex" of smaller dimension, and to compensate for this reduction by allowing vertices to repeatedly app…

由几何图导出的相交直径球 II

Intersecting diametral balls induced by a geometric graph II

本文证明了任意有限点集的 max-sum tree 均为 Tverberg 图，推广了平面情形的结论，并为偶数点集的 max-sum matching 是 Tverberg 图提供了新证明及更强版本。

由最大和匹配导出的相交椭圆

Intersecting ellipses induced by a max-sum matching

本文研究了平面偶数点集最大和匹配中由特定椭圆围成的凸集相交问题，通过优化方法证明了这些凸集必然相交，从而解决了1995年提出的Tverberg型问题。

$d$-可坍缩与$d$-Leray复形的彩色Helly定理的推广

Extensions of the Colorful Helly Theorem for $d$-collapsible and $d$-Leray complexes

本文推广了$d$-可坍缩和$d$-Leray复形的彩色Helly定理，统一了多个现有版本，并以此推导出Tverberg定理的扩展形式。