Related papers: On the sunflower bound for $k$-spaces, pairwise in…

We introduce the concepts of complex Grassmannian codes and designs. Let G(m,n) denote the set of m-dimensional subspaces of C^n: then a code is a finite subset of G(m,n) in which few distances occur, while a design is a finite subset of…

Let $X$ be an $n$-element point set in the $k$-dimensional unit cube $[0,1]^k$ where $k \geq 2$. According to an old result of Bollob\'as and Meir (1992), there exists a cycle (tour) $x_1, x_2, \ldots, x_n$ through the $n$ points, such that…

A topological graph is \emph{$k$-quasi-planar} if it does not contain $k$ pairwise crossing edges. A topological graph is \emph{simple} if every pair of its edges intersect at most once (either at a vertex or at their intersection). In…

We enumerate the number of $T$-splitting subspaces of dimension $m$ for an arbitrary operator $T$ on a $2m$-dimensional vector space over a finite field. When $T$ is regular split semisimple, comparison with an alternate method of…

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest…

A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$…

It is proved that for $k\geq 4$, if the points of $k$-dimensional Euclidean space are coloured in red and blue, then there are either two red points distance one apart or $k+3$ blue collinear points with distance one between any two…

Using the largest principal angle as a distance between same-dimensional linear subspaces of $\mathbb{R}^n$, we construct $k$-dimensional subspaces which deviate from every coordinate $k$-subspace by at least $\arccos(1/\sqrt n)$. The…

Let $n$ and $t$ be positive integers with $t<n$, and let $q$ be a prime power. A partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of ${\rm PG}(n-1,q)$ that are pairwise disjoint. Let $r\equiv n\pmod{t}$…

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function $f(r)$ such that, for any set $C$ of $r$-state characters, $C$ is compatible if and only…

A graph $G$ is {$k$-crossing-critical} if $cr(G)\ge k$, but $cr(G\setminus e)<k$ for each edge $e\in E(G)$, where $cr(G)$ is the crossing number of $G$. It is known that for any $k$-crossing-critical graph $G$, $cr(G)\le 2.5k+16$ holds, and…

We give a complete description of the set of triples (a,b,c) of real numbers with the following property. There exists a constant K such that a n_3 + b n_2 + c n_1 - K is a lower bound for the matching number of every connected subcubic…

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations \pi, \sigma in S there is a point i in {1,...,n} such that \pi(i)=\sigma(i). Deza and Frankl \cite{MR0439648} proved that if S a…

The maximum possible number of non-overlapping unit spheres that can touch a unit sphere in $n$ dimensions is called kissing number. The problem for finding kissing numbers is closely connected to the more general problems of finding bounds…

A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological…

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…

Finding problems that allow for superpolynomial quantum speedup is one of the most important tasks in quantum computation. A key challenge is identifying problem structures that can only be exploited by quantum mechanics. In this paper, we…

For a code $C$ in a space with maximal distance $n$, we say that $C$ has symmetric distances if its distance set $S(C)$ is symmetric with respect to $n / 2$. In this paper, we prove that if $C$ is a binary code with length $2n$, constant…