Related papers: Erd\H{o}s--Szekeres-type problems in the real proj…
The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of $r$ mutually orthogonal Heffter systems for any…
The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mapping in Hausdorff p-vector spaces, and the fixed point theorem for upper semicontinuous set-valued mappings in Hausdorff locally…
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…
We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d)<d$ such…
Erd\H{o}s and Guy initiated a line of research studying $\mu_k(n)$, the minimum number of convex $k$-gons one can obtain by placing $n$ points in the plane without any three of them being collinear. Asymptotically, the limits $c_k :=…
A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a…
According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log…
In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds…
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 $n$…
A $(v,k;r)$ Heffter space is a resolvable $(v_r,b_k)$ configuration whose points form a half-set of an abelian group $G$ and whose blocks are all zero-sum in $G$. It was recently proved that there are infinitely many orders $v$ for which,…
Satisfiability solving has been used to tackle a range of long-standing open math problems in recent years. We add another success by solving a geometry problem that originated a century ago. In the 1930s, Esther Klein's exploration of…
The conjecture of Brown, Erd\H{o}s and S\'os from 1973 states that, for any $k \ge 3$, if a $3$-uniform hypergraph $H$ with $n$ vertices does not contain a set of $k+3$ vertices spanning at least $k$ edges then it has $o(n^2)$ edges. The…
A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of…
Erd\H{o}s and Purdy, and later Agarwal and Sharir, conjectured that any set of $n$ points in $\mathbb R^{d}$ determine at most $Cn^{d/2}$ congruent $k$-simplices for even $d$. We obtain the first significant progress towards this…
We prove that if $N$ points lie in convex position in the plane then they determine $\Omega(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in…
In this work, we consider a class of convex optimization problems in a real Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those…
Given an algebraically closed field $K$ of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of $\mathbb P^2(K)$. Answering…
The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior.…
We describe in this paper a geometric construction in the projective p-adic plane that gives, together with a suitable notion of p-adic convexity, some open subsets of P 2 .Q p / naturally endowed with a "Hilbert" distance and a transitive…
For self-gravitating, static, spherically symmetric, minimally coupled scalar fields with arbitrary potentials and negative kinetic energy (favored by the cosmological observations), we give a classification of possible regular solutions to…