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We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was…

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point…

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…

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…

We revisit classic balancing problems for linear extensions of a partially ordered set $P$, proving results that go far beyond many of the best earlier results on this topic. For example, with $p(x\prec y)$ the probability that $x$ precedes…

Let $S$ be a point set in the plane such that each of its elements is colored either red or blue. A matching of $S$ with rectangles is any set of pairwise-disjoint axis-aligned rectangles such that each rectangle contains exactly two points…

For $0<q\le 2,\ 1\le k < n,$ let $X=(X_1,...,X_n)$ and $Y=(Y_1,...,Y_n)$ be symmetric $q$-stable random vectors so that the joint distributions of $X_1,...,X_k$ and $X_{k+1},...,X_n$ are equal to the joint distributions of $Y_1,...,Y_k$ and…

Given $n>0$, let $S\subset [0,1]^2$ be a set of $n$ points, chosen uniformly at random. Let $R\cup B$ be a random partition, or coloring, of $S$ in which each point of $S$ is included in $R$ uniformly at random with probability $1/2$.…

We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through…

We consider an incidence problem in $\mathbb{R}^4$ which asks, for a set of $L$ lines and a set of $S$ planes in general position, what the maximum number of line-plane incidences is. A line-plane incidence is defined as a point where a…

Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…

We introduce the $(p,q)$-Fair Clustering problem. In this problem, we are given a set of points $P$ and a collection of different weight functions $W$. We would like to find a clustering which minimizes the $\ell_q$-norm of the vector over…

In this paper, we continue the study of linear sets with complementary weights. We find criteria to determine the set of points of any fixed weight and use this to present particular linear sets with few points of weight more than one. We…

We investigate dense lineability and spaceability of subsets of $\ell_\infty$ with a prescribed number of accumulation points. We prove that the set of all bounded sequences with exactly countably many accumulation points is densely…

Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the…

Suppose we are given a set $\cal B$ of blue points and a set $\cal R$ of red points, all lying above a horizontal line $\ell$, in the plane. Let the weight of a given point $p_i\in {\cal B}\cup{\cal R}$ be $w_i>0$ if $p_i\in {\cal B}$ and…

Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum…

Consider the projections of a finite set $A\subset R^n$ onto the coordinate hyperplanes. How small can the sum of the sizes of these projections be, given the size of $A$? In a different form, this problem has been studied earlier in the…

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…

We consider the following problem: Let $\mathcal{L}$ be an arrangement of $n$ lines in $\mathbb{R}^3$ colored red, green, and blue. Does there exist a vertical plane $P$ such that a line on $P$ simultaneously bisects all three classes of…