A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springe

Eliminating higher-multiplicity intersections in the metastable dimension range

The $r$-fold analogues of Whitney trick were `in the air' since 1960s. However, only in this century they were stated, proved and applied to obtain interesting results. Here we prove and apply a version of the $r$-fold Whitney trick when general position $r$-tuple intersections have positive dimension. Theorem. Assume that $D=D_1\sqcup\ldots\sqcup D_r$ is disjoint union of $k$-dimensional disks, $rd\ge (r+1)k+3$, and $f:D\to B^d$ a proper PL (smooth) map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$. If the map $$f^r:\partial(D_1\times\ldots\times D_r)\to (B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ |\ x\in B^d\}$$ extends to $D_1\times\ldots\times D_r$, then there is a proper PL (smooth) map $\overline f:D\to B^d$ such that $\overline f=f$ on $\partial D$ and $\overline fD_1\cap\ldots\cap \overline fD_r=\emptyset$.Comment: 13 pages, 2 figures, exposition improve

Stronger counterexamples to the topological Tverberg conjecture

Denote by $\Delta_N$ the $N$-dimensional simplex. A map $f\colon \Delta_N\to\mathbb R^d$ is an almost $r$-embedding if $f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if $r$ is not a prime power and $d\ge2r+1$, then there is an almost $r$-embedding $\Delta_{(d+1)(r-1)}\to\mathbb R^d$. This was improved by Blagojevi\'c-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking $k$-fold join power of lower-dimensional ones. We improve this further (for $d$ large compared to $r$): If $r$ is not a prime power and $N:=(d+1)r-r\Big\lceil\dfrac{d+2}{r+1}\Big\rceil-2$, then there is an almost $r$-embedding $\Delta_N\to\mathbb R^d$. For the $r$-fold van Kampen-Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard-Wagner theorem on construction of almost $r$-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.Comment: 7 page

LIPIcs

The Tverberg 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 R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,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 concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2