Related papers: Many antipodes implies many neighbors

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to…

We prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^(r) are adjacent if |x \Delta y|=2. Namely, if \mathcal{A} \subset [n]^(r) with |\mathcal{A}|=\alpha {n \choose r},…

Let $q,d\geq 2$ be integers. Define $$ J(q,d):=\frac 1q \Big( \min_{0<x<1} \frac{1-x^q}{1-x} x^{-\frac{q-1}{d}}\Big). $$ Let $\mbox{$\cal G$}\subseteq {\mathbb R}^n$ be an arbitrary subset. We denote by $d(\mbox{$\cal G$})$ the set of…

Given a matching $M$ in the hypercube $Q^n$, the \emph{profile} of $M$ is the vector $\boldsymbol{x}=(x_1,\ldots, x_n) \in \mathbb{N}^n$ such that $M$ contains $x_i$ edges whose endpoints differ in the $i$th coordinate. If $M$ is a perfect…

Felsner, Li and Trotter showed that the dimension of the adjacency poset of an outerplanar graph is at most 5, and gave an example of an outerplanar graph whose adjacency poset has dimension 4. We improve their upper bound to 4, which is…

Given a set $X\subseteq\mathbb{R}^2$ of $n$ points and a distance $d>0$, the multiplicity of $d$ is the number of times the distance $d$ appears between points in $X$. Let $a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$ denote the…

A poset $\mathbf{P} = (X,\preceq)$ is {\em $m$-partite} if $X$ has a partition $X = X_1 \cup ... \cup X_m$ such that (1) each $X_i$ forms an antichain in $\mathbf{P}$, and (2) $x\prec y$ implies $x\in X_i$ and $y\in X_j$ where $i<j$. In…

We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of $C^\infty$ norms on $\R^3$ admitting six equidistant points, which refutes a conjecture of…

We show that multiple orthogonal polynomials for r measures $(\mu_1,...,\mu_r)$ satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices $\vec{n}\pm \vec{e}_j$, where $\vec{e}_j$ are the standard unit…

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…

We prove that if $P$ is a set of $n$ points in $\mathbb{C}^2$, then either the points in $P$ determine $\Omega(n^{1-\epsilon})$ complex distances, or $P$ is contained in a line with slope $\pm i$. If the latter occurs then each pair of…

We prove that every $n$ vertex linear triple system with $m$ edges has at least $m^6/n^7$ copies of a pentagon, provided $m>100 \, n^{3/2}$. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More…

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability…

If $a_1, a_2, ..., a_k$ and $n$ are positive integers such that $n = a_1 + a_2 + ... + a_k$, then the sum $a_1 + a_2 + ... + a_k$ is said to be a \emph{partition of $n$} of \emph{length $k$}, and $a_1, a_2, ..., a_k$ are said to be the…

For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair…

We show that any total preorder on a set with $\binom{n}{2}$ elements coincides with the order on pairwise distances of some point collection of size $n$ in $\mathbb{R}^{n-1}$. For linear orders, a collection of $n$ points in…

For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted'…

We show that if the Banach-Mazur distance between an n-dimensional normed space X and ell infinity is at most 3/2, then there exist n+1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary…

We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the…

Two members of a packing are neighbors if they have a common boundary point. A multitude of problems arises in connection with neighbors in a packing. The oldest one concerns a dispute between Newton and Gregory about the maximum number of…