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Register a new userRegular Polygonal Partitions of a Tverberg Type
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A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in $mathbb{R}^d$ cannot be so divided, and in these cases we ask if the set can nonetheless be $P(r,d)$--partitioned, i.e., split into $r$ subsets so that there exist $r$ points, one from each resulting convex hull, which form the vertex set of a prescribed convex $d$--polytope $P(r,d)$. Our main theorem shows that this is the case for any generic $T(r,2)-2$ points in the plane and any $rgeq 3$ when $P(r,2)=P_r$ is a regular $r$--gon, and moreover that $T(r,2)-2$ is tight. For higher dimensional polytopes and $r=r_1cdots r_k$, $r_i geq 3$, this generalizes to $T(r,2k)-2k$ generic points in $mathbb{R}^{2k}$ and orthogonal products $P(r,2k)=P_{r_1}times cdots times P_{r_k}$ of regular polygons, and likewise to $T(2r,2k+1)-(2k+1)$ points in $mathbb{R}^{2k+1}$ and the product polytopes $P(2r,2k+1)=P_{r_1}times cdots times P_{r_k} times P_2$. As with Tverbergs original theorem, our results admit topological generalizations when $r$ is a prime power, and, using the constraint method of Blagojevic, Frick, and Ziegler, allow for dimensionally restrict
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For a graph whose vertex set is a finite set of points in $mathbb R^d$, 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. Using the idea of infinite descent, we prove that (iii) for any even set of points in $mathbb R^d$, there exists a perfect matching that is an open Tverberg graph; (iv) for any $n$ red and $n$ blue points in $ mathbb R^d $, there exists a perfect red-blue matching that is a Tverberg graph.
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 Tverbergs famous 1966 generalization of the classical Radons Theorem. Although established for all prime powers in 1987 by Ozaydin, 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. Starting with a reformulation of the topological Tverberg conjecture in terms of harmonic analysis on finite groups, we show that despite the failure of the conjecture, continuous maps textit{below} the tight dimension $N(q,d)$ are nonetheless guaranteed $q$ pairwise disjoint subfaces -- including when $q$ is not a prime power -- which satisfy a variety of average value coincidences, the latter obtained as the vanishing of prescribed Fourier transforms.
We present a motivated exposition of the proof of the following Tverberg Theorem: For every integers $d,r$ any $(d+1)(r-1)+1$ points in $mathbb R^d$ can be decomposed into $r$ groups such that all the $r$ convex hulls of the groups have a common point. The proof is by well-known reduction to the Barany Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding $mathbb R^dsubsetmathbb R^{d+1}$, of vectors $varphi_{j,i}$ and statement of the Barany Theorem) are not introduced in advance in a non-motivated way, but naturally appear in an attempt to construct the required decomposition. This attempt is based on rewriting several equalities between vectors as one equality between vectors of higher dimension.
Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $Sigma$ such that every continuous map $fcolon Sigma to mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in $mathbb{R}^d$. Such results are plentiful for $q$ a power of a prime. However, for $q$ with at least two distinct prime divisors, results that guarantee the existence of $q$-fold points of coincidence are non-existent---aside from immediate corollaries of the prime power case. Here we present a general method that yields such results beyond the case of prime powers. In particular, we prove previously conjectured upper bounds for the topological Tverberg problem for all $q$.
Szemeredis Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as gentle classes.
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