Related papers: Exponentially sized pointsets with angles less tha…

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2…

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position…

In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$ as a function of the maximum angle $\angle A$ that any three of these points form, provided $\angle A < \theta_d := \arccos(-\frac 1 {d}) \in…

We prove that, for any $B \subset \mathbb R$, the Cartesian product set $B \times B$ determines $\Omega(|B|^{2+c})$ distinct angles.

Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the…

Let $S$ be a set of $n$ points in the plane, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that…

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…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer $d$ there is a constant $c_d > 0$ such that whenever $X_1,..., X_{d+1}$ are $n$-element subsets of $\mathbb{R}^d$, then we…

The Danzer--Gr\"unbaum acute angles problem asks for the largest size of a set of points in ${\mathbb R}^d$ that determines only acute angles. Recently, the problem was essentially solved thanks to the results of the second author and of…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…

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…

Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew…

For a positive integer $d$, a set of points in $d$-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let $f(d)$ denote the largest size of an almost-equidistant set…

We present a simple construction of an acute set of size $2^{d-1}+1$ in $\mathbb{R}^d$ for any dimension $d$. That is, we explicitly give $2^{d-1}+1$ points in the $d$-dimensional Euclidean space with the property that any three points form…

We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many…

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) $$\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].$$ This is tight up to…

Let OT_d(n) be the smallest integer N such that every N-element point sequence in R^d in general position contains an order-type homogeneous subset of size n, where a set is order-type homogeneous if all (d+1)-tuples from this set have the…

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

Given positive integers $k\leq d$ and a finite field $\mathbb{F}$, a set $S\subset\mathbb{F}^{d}$ is $(k,c)$-subspace evasive if every $k$-dimensional affine subspace contains at most $c$ elements of $S$. By a simple averaging argument, the…