Related papers: Codimension two and three Kneser Transversals

A $k$-plane of a $d$-dimensional array is a subarray formed by fixing $d-k$ coordinates and allowing the remaining $k$ coordinates to vary freely. A Latin hypercube of dimension $d$ and order $n$ is an $n\times n\times\cdots\times n$ array…

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

For integers $d \geq 2$ and $k \geq d+1$, a $k$-hole in a set $S$ of points in general position in $\mathbb{R}^d$ is a $k$-tuple of points from $S$ in convex position such that the interior of their convex hull does not contain any point…

We establish a transversality theorem for multiple-point crossings under generic linear perturbations with explicit Hausdorff measure estimates for the exceptional parameter set, and hence explicit upper bounds on its Hausdorff dimension.…

For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $\delta(E)$. We show that for any fixed $k$-point pattern $V$, there is a set…

The quantum Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for 2nd order superintegrable…

Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ \lambda_1 s_1 + \cdots + \lambda_k s_k : \lambda_i \in [0,1], \sum_{i=1}^k \lambda_i = 1 , s_i \in A \right\} \end{align*} to be the set of vectors…

We demonstrate the existence and stability of one-dimensional (1D) topological kink configurations immersed in higher-dimensional bosonic gases and nonlinear optical setups. Our analysis pertains, in particular, to the two- and…

We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{\Bbb{N}_0})$ in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one…

We consider the continuous $W^{2,d}$ immersions of $d$-dimensional hypersurfaces in $\mathbb{R}^{d+1}$ with second fundamental forms uniformly bounded in $L^d$. Two results are obtained: first, a family of such immersions is constructed,…

For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $\delta\in (0,1)$, the $\delta$-neighbourhood of $K$ has Lebesgue measure $\lesssim…

Let us say that an element of a given family $\A$ of subsets of $\R^d$ can be reconstructed using $n$ test sets if there exist $T_1,...,T_n \subset \R^d$ such that whenever $A,B\in \A$ and the Lebesgue measures of $A \cap T_i$ and $B \cap…

Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to…

The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any…

Let X be a real normed vector space and dim X \ge 2. Let d>0 be a fixed real number. We prove that if x,y \in X and ||x-y||/d is a rational number then there exists a finite set {x,y} \subseteq S(x,y) \subseteq X with the following…

Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if…

Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos--Szekeres theorem asserts that every such P contains a monotone subsequence S of $\sqrt N$ points. Another, equally…

We study the transverse spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation. We show that these waves are transversely unstable with…

In this paper, we prove that if $S\subseteq\mathbb{R}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq\mathbb{R}^d$, then $\mathrm{dim}_H S\cap K=\mathrm{dim}_H K$ for any irreducible self-conformal set…

The matching number of a $k$-graph is the maximum number of pairwise disjoint edges in it. The $k$-graph is called $t$-resilient if omitting $t$ vertices never decreases its matching number. The complete $k$-graph on $sk+k-1$ vertices has…