Related papers: Complete Kneser Transversals

The dichromatic number of a digraph $D$ is the smallest $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations.…

Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying \begin{align*} \left|\sum_{j=1}^{n}\lambda_j^2\right|=\max_{1\leq j \leq n}|\lambda_j|^2, \quad \forall \lambda_j \in \mathbb{K}, 1\leq j \leq n, \forall n \in…

An $(n,d,\lambda)$-graph is a $d$ regular graph on $n$ vertices in which the absolute value of any nontrivial eigenvalue is at most $\lambda$. For any constant $d \geq 3$, $\epsilon>0$ and all sufficiently large $n$ we show that there is a…

We consider four problems. Rogers proved that for any convex body $K$, we can cover ${\mathbb R}^d$ by translates of $K$ of density very roughly $d\ln d$. First, we extend this result by showing that, if we are given a family of positive…

Kneser-Haken Finiteness asserts that for each compact 3-manifold M there is an integer c(M) such that any collection of k>c(M) closed, essential, 2-sided surfaces in M must contain parallel elements. We show here that if M is closed then…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…

For positive integers $k < n$ such that $k$ divides $n$, let $(n)^k_{\hom}$ be the set of homogeneous $k$-partitions of $\{1, \dots, n\}$, that is, the set of partitions of $\{1, \dots, n\}$ into $k$ classes of the same cardinality. In the…

Let $n, m, k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n,…

Let A be an m \times n matrix in which the entries of each row are all distinct. Drisko showed that, if m \ge 2n-1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices…

For each pair (k,r) of positive integers with r>1, we consider an ideal I^(k,r)_n of the ring of symmetric polynomials in n variables. The ideal I_n^(k,r) has a basis consisting of Macdonald polynomials P(x_1,...,x_n;q,t) at…

Let $K$ be a proper (i.e., closed, pointed, full convex) cone in ${\Bbb R}^n$. An $n\times n$ matrix $A$ is said to be $K$-primitive if there exists a positive integer $k$ such that $A^k(K \setminus \{0 \}) \subseteq$ int $K$; the least…

We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k.…

We solve a long-standing open problem posed by Goodman \& Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal for any $0 \le k \le d-1$. This result…

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with…

In this paper, we study the $d$-dimensional rectilinear drawings of the complete $d$-uniform hypergraph $K_{2d}^d$. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove…

We give new proofs of the description convex hulls of space curves $\gamma : [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(\gamma',…

Let $d\geq 2$ be an integer, $K_0$ a perfect field such that $char(K_0)$ does not divide $d$, $n > d$ an integer prime to $d$, $f(x)\in K_0[x]$ a degree $n$ monic polynomial without repeated roots, and $C_{f,d}$ a smooth projective model of…

Let $K$ be a maximal lattice-free set in $\mathbb{R}^d$, that is, $K$ is convex and closed subset of $\mathbb{R}^d$, the interior of $K$ does not cointain points of $\mathbb{Z}^d$ and $K$ is inclusion-maximal with respect to the above…

We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in…