Let $P$ be a set $n$ points in a $d$-dimensional space. Tverberg's theorem says that, if $n$ is at least $(k-1)(d+1)+1$, then $P$ can be partitioned into $k$ sets whose convex hulls intersect. Partitions with this property are called {\em Tverberg partitions}. A partition has tolerance $t$ if the partition remains a Tverberg partition after removal of any set of $t$ points from $P$. Tolerant Tverberg partitions exist in any dimension provided that $n$ is sufficiently large. Let $N(d,k,t)$ be the smallest value of $n$ such that tolerant Tverberg partitions exist for any set of $n$ points in $\mathbb{R}^d$. Only few exact values of $N(d,k,t)$ are known.
In this paper we establish a new tight bound for $N(2,2,2)$. We also prove many new lower bounds on $N(2,k,t)$ for $k\ge 2$ and $t\ge 1$.
In 2009, Blagojevic, Matschke & Ziegler established the first tight colored Tverberg theorem, but no lower bounds for the number of colored Tverberg partitions. We develop a colored version of our previous results (2008), and we extend our results from the uncolored version: Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. This follows from similar results on the number of colored Birch partitions.
We give an extension of the lower bound of Vucic and Zivaljevic for the number of Tverberg partitions from the prime to the prime power case. Our proof is inspired by the Z_p-index version of the proof in Matousek's book "Using the Borsuk-Ulam Theorem" and uses Volovikov's Lemma. Analogously, one obtains an extension of the lower bound for the number of different splittings of a generic necklace to the prime power case.
A theorem of Tverberg from 1966 asserts that every set $X\subset\mathbb{R}^d$ of $n=T(d,r)=(d+1)(r-1)+1$ points can be partitioned into $r$ pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of $n$ into $r$ parts (that is, $r$ integers $a_1,\ldots,a_r$ satisfying $n=a_1+\cdots+a_r$), where the parts $a_i$ correspond to the number of points in every subset. In this paper, we prove that for any partition $a_i\le d+1$, $i=1,\ldots,r$, there exists a set $X\subset\mathbb{R}^d$ of $n$ points, such that every Tverberg partitio…
Tverberg's theorem bounds the number of points $\mathbb{R}^d$ needed for the existence of a partition into $r$ parts whose convex hulls intersect. If the points are colored with $N$ colors, we seek partitions where each part has at most one point of each color. In this manuscript, we bound the number of color classes needed for the existence of partitions where the convex hulls of the parts intersect even after any set of $t$ colors is removed. We prove asymptotically optimal bounds for $t$ when $r \le d+1$, improve known bounds when $r>d+1$, and give a geometric characterization for the confi…
A result of Rosenthal says that for every $q>1$ and $n \in \mathbb{N}$ there is $N \in \mathbb{N}$ such that every sequence of $N$ distinct positive numbers contains, after a suitable translation and possible multiplication by $-1$, a subsequence $a_1,\ldots,a_n$ that is either $q$-increasing (that is, $a_{i+1}>qa_i$ for all $i$) or $1/q$-decreasing ($a_{i+1}<a_i/q$ for all $i$). One of our main theorems extends this result to vector sequences. This theorem is then used to prove the universality theorem for Tverberg partitions which says that, for every $d$ and $r$, every long enough sequence …
In this paper we give an asymptotically tight bound for the tolerated Tverberg Theorem when the dimension and the size of the partition are fixed. To achieve this we study certain partitions of order-type homogeneous sets and use a generalization of the Erdős-Szekeres theorem.
We prove a Tverberg-type theorem using the probabilistic method. Given $\varepsilon >0$, we find the smallest number of partitions of a set $X$ in $R^d$ into $r$ parts needed in order to induce at least one Tverberg partition on every subset of $X$ with at least $\varepsilon |X|$ elements. This generalizes known results about Tverberg's theorem with tolerance.
The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have non-empty $q$-fold intersection. The affine cases, true for all $q$, constitute Tverberg's famous 1966 generalization of the classical Radon's Theorem. Although established for all prime powers in 1987 by Özaydin, counterexamples to the conjecture, relying on 2014 work of Mabillard and Wagner, were first shown to exist for all non-prime-powers in 2015 by Frick.…
Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal I$ "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition. We say that $\mathcal I$ is "unavoidable" if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete charact…
In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set.
Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness, and a lower bound. This implies the first non-trivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments, and do not use the equivariant method from topological combinatorics.
In 2008 Karasev conjectured that for every set of $r$ blue lines, $r$ green lines, and $r$ red lines in the plane, there exists a partition of them into $r$ colorful triples whose induced triangles intersect. We disprove this conjecture for every $r$ and extend the counterexamples to higher dimensions.
The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d + 1)(r - 1) + 1 are known as Tverberg numbers. Reay asks the following question: if we add an additional parameter k (1 < k < r+1) what is the minimal number of points we need in order to guarantee that there exists an r partition of them such that any k of the r convex hulls intersect. This minimal number is denoted by T(d,r,k). Reay conjectured that T(d,r,k) = T(d,r) for all d,r and k. In this a…
The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity r…
We provide an algorithm that verifies the optimal colored Tverberg problem for $10$ points in the plane: Every $10$ points in the plane in color classes of size at most $3$ can be partitioned in $4$ rainbow pieces such that their convex hulls intersect in a common point. This is achieved by translating the problem to $k$-partite graphs and using a new algorithm to verify that those graphs do not have a $k$-clique.
We use the probabilistic method to obtain versions of the colorful Carathéodory theorem and Tverberg's theorem with tolerance. In particular, we give bounds for the smallest integer $N=N(t,d,r)$ such that for any $N$ points in $R^d$, there is a partition of them into $r$ parts for which the following condition holds: after removing any $t$ points from the set, the convex hulls of what is left in each part intersect. We prove the bound $N=rt+O(\sqrt{t})$ for fixed $r,d$ which is polynomial in each parameters. Our bounds extend to colorful versions of Tverberg's theorem, as well as Reay-type var…
We present projective versions of the center point theorem and Tverberg's theorem, interpolating between the original and the so-called "dual" center point and Tverberg theorems. Furthermore we give a common generalization of these and many other known (transversal, constraint, dual, and colorful) Tverberg type results in a single theorem, as well as some essentially new results about partitioning measures in projective space.
Tverberg's theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg's theorem is but a special case of a more general situation. Given sufficiently many points, all trees and cycles can also be induced by at least one partition of a point set.
Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction of Tverberg's theorem from the special case k=2. We dash the hopes of a purely combinatorial deduction, but show that the case k=2 does imply that every set of O(k^2 log^2 k) points admits a Tverberg partition into k parts.
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are…
The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in R^d of cardinality (d+1)t, partitioned into t-point subsets C_1,C_2,...,C_{d+1} (which we think of as color classes; e.g., the points of C_1 are red, the points of C_2 blue, etc.), there exist r disjoint sets R_1,R_2,...,R_r \subseteq C that are "rainbow", meaning that |R_i \cap C_j| < 2 for every i,j, and whose convex hulls all have a common point. All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and…
This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carathéodory's theorems.
Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concr…
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 comb…
Tverberg's theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. In arXiv:1808.00551v1 [math.MG] the authors investigate how other simplicial complexes arise as nerve complexes once we have a set with sufficiently many points. In this paper we relate the theory of word-representable graphs as a way of codifying $1$-skeletons of simplicial complexes to generate nerves. In particular, we show that every $2$-word-representable triangle-free g…
We prove the following optimal colorful Tverberg-Vrecica type transversal theorem: For prime r and for any k+1 colored collections of points C^l of size |C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k) is even or k=0. Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0…
