Related papers: Large grid subsets without many cospherical points

Erd\H{o}s asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? F\"uredi constructed a set of $n$…

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…

In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any $n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They…

Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered…

In 2008, Bukh, Matousek, and Nivasch conjectured that for every n-point set S in R^d and every k, 0 <= k <= d-1, there exists a k-flat f in R^d (a "centerflat") that lies at "depth" (k+1) n / (k+d+1) - O(1) in S, in the sense that every…

Let $f(n,v,e)$ denote the maximum number of edges in a $3$-uniform hypergraph not containing $e$ edges spanned by at most $v$ vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of…

Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We show that if…

We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$, improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.

In 1935, Erd\H{o}s and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a…

Boros and Furedi (for d=2) and Barany (for abritrary d) proved that there exists a positive real number c_d such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d…

We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be…

A set N is called a "weak epsilon-net" (with respect to convex sets) for a finite set X in R^d if N intersects every convex set that contains at least epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets X in R^d…

We show quantum lower bounds for two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. It has been known that, for any $k$,…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point…

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…

According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$…

The directed graph reachability problem takes as input an $n$-vertex directed graph $G=(V,E)$, and two distinguished vertices $s$ and $t$. The problem is to determine whether there exists a path from $s$ to $t$ in $G$. This is a canonical…

A set of integers greater than 1 is primitive if no element divides another. Erd\H{o}s proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked…

Motivated by classical work of Alon and F\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) :=…