Optimal bounds for the colored Tverberg problem

Blagojević, Pavle V. M.; Matschke, Benjamin; Ziegler, Günter M.

doi:10.4171/jems/516

Cited by 59 publications

(54 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If k + 1 is prime, this was solved by Blagojević, Matschke and Ziegler [4] with topological methods (equivariant obstruction theory). At the core of their argument is the computation of degrees for a simplicial pseudomanifold; this was made explicit by Vrećica andŽivaljević [23].…”

Section: Introductionmentioning

confidence: 97%

Equal coefficients and tolerance in coloured tverberg partitions

Soberón

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi [2] and asks whether for any d + 1 sets (considered as colour classes) of k points each in R d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d = 1, 2 [3] or k + 1 is prime [5]. In this paper we show that (k − 1)d + 1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes. Namely, if we have (r + 1)(k − 1)d + 1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed. We also investigate the relation of the case k = 2 and the Gale transform. We then show applications of these results to purely combinatorial problems.

show abstract

Section: Introductionmentioning

confidence: 97%

Equal coefficients and tolerance in coloured tverberg partitions

Soberón

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

show abstract

“…It is known that any such map necessarily has an r-Tverberg point if m = d + 1, each |Vi| has size at least 2r − 1, and r is a prime or a prime power [39,34]. In a recent breakthrough [4], sharp bounds were obtained for the case that r + 1 is a prime: in this case, it is sufficient to have…”

Section: Introductionmentioning

confidence: 99%

“…Throughout, we work under the assumptions (4). The proof uses a number of standard notions and techniques from PL topology, for which we refer to [24].…”

Section: The R-fold Whitney Trickmentioning

confidence: 99%

See 1 more Smart Citation

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Mabillard

Wagner

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) selfintersection points of maps from finite simplicial complexes (compact polyhedra) into R d and study conditions under which such multiple points can be eliminated.The most classical case is that of embeddings (i.e., maps without double points) of a k-dimensional complex K into R 2k . For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen obstruction). For k ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f : K → R 2k , which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f . One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign.We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y ∈ R d an r-fold Tverberg point of a map f :The analogue of (non-)embeddability that we study is the problem Tverberg r k→d : Given a k-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f :Tverberg point? Here, we show that for fixed r, k and d of the form d = rm and k = (r − 1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings).Our main tool is an r-fold analogue of the Whitney trick: * Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org. SoCG'14, June 8-11, 2014, Kyoto, Japan. Copyright is held by the owner/author(s). Publication rights licensed to ACM. ACM 978-1-4503-2594-3/14/06 ...$15.00.Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y− of opposite intersection sign, we can eliminate y+ and y− by a local isotopy of f .In a subsequent paper, we plan to develop this further and present a generalization of the classical Haefliger-Weber Theorem (which yields a necessary and sufficient condition for embeddability of k-complexes into R d for a wider range of dimensions) to intersection points of higher multiplicity.

show abstract

“…(4) The topological Tverberg theorem has an "optimal" colored extension [4] that has the color-free original version as a special case: Note for this the concepts of "coloring" and "rainbow faces" have been redefined a bit: we use more than + 1 colors, and the rainbow faces do not pick up all colors. Among all the variations of the topological Tverberg theorem, this theorem is the only one that extends it, except that we proved it (together with Benjamin Matschke) only for the case where is prime, even in the case where the map is affine.…”

Section: The Next Fifty Years In the Life Of Tverberg's Theorem?mentioning

confidence: 99%