We show that the 5-regular matchstick graphs on the sphere are exactly the five 5-regular contact graphs of congruent caps on the sphere found by R. M. Robinson (1969).

Let P be a set of n points in the plane, not all on a line, each colored red or blue. The classical Motzkin–Rabin theorem guarantees the existence of a monochromatic line. Motivated by the seminal work of Green and Tao (2013) on the Sylvester-Gallai theorem, we investigate the quantitative and structural properties of monochromatic geometric objects, such as lines, circles, and conics. We first show that if no line contains more than three points, then for all sufficiently large n there are at least n^2/24 - O(1) monochromatic lines. We then show a converse of a theorem of Jamison (1986): Given n≥ 6 blue points and n red points, if the blue points lie on a conic and every line through two blue points contains a red point, then all red points are collinear. We also settle the smallest nontrivial case of a conjecture of Milićević (2018) by showing that if we have 5 blue points with no three collinear and 5 red points, if the blue points lie on a conic and every line through two blue points contains a red point, then all 10 points lie on a cubic curve. Further, we analyze the random setting and show that, for any non-collinear set of n≥ 10 points independently colored red or blue, the expected number of monochromatic lines is minimized by the near-pencil configuration. Finally, we examine monochromatic circles and conics, and exhibit several natural families in which no such monochromatic objects exist.

We prove that there are arbitrarily large equilateral sets of planar and symmetric convex bodies in the Banach–Mazur distance. The order of the size of these d d -equilateral sets asymptotically matches the bounds of the size of maximum-size d d -separated sets (determined by Bronšteĭn in 1978), showing that our construction is essentially optimal.

A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a matchstick graph. We prove two somewhat counterintuitive results concerning the maximum number of edges of such graphs in two different scenarios. First, we show that there is a constant c>0 such that every triangle-free matchstick graph on n vertices has at most 2n-c√(n) edges. This statement is not true for any c>√(2). We also prove that for every r>0, there is a constant ε(r)>0 with the property that every matchstick graph on n vertices contained in a disk of radius r has at most (2-ε(r))n edges.

. A matchstick graph is a crossing-free unit-distance graph in the plane. Harborth (1981) proposed the problem of determining whether there exists a matchstick graph in which every vertex has degree exactly 5. In 1982, Blokhuis gave a proof of non-existence. A shorter proof was found by Kurz and Pinchasi (2011) using a discharging method. We combine their method with the isoperimetric inequality to show that there are Ω( √ n ) vertices in a matchstick graph on n vertices that are of degree at most 4, which is asymptotically tight.

We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper bound for the number of edges in terms of $n$ and the number of non-triangular faces. We also find a sharp upper bound for the number of triangular faces in a matchstick graph.

We study the contact structure of totally separable packings of translates of a convex body K in ℝ^d, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number H_sep(K) of K is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of K. We show that for each d≥ 8, there exists a smooth and strictly convex K in ℝ^d with H_sep(K)>2d, and asymptotically, H_sep(K)=Ω((3/√(8))^d). We show that Alon's packing of Euclidean unit balls such that each translate touches at least 2^√(d) others whenever d is a power of 4, can be adapted to give a totally separable packing of translates of the ℓ_1-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of n translates of any planar convex body K. We prove that the maximum equals ⌊ 2n-2√(n)⌋ if and only if K is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.

We resurrect an old definition of the linear measure of a metric continuum in terms of Steiner trees, independently due to Menger (1930) and Choquet (1938). We generalise it to any metric space and provide a proof of a little-known theorem of Choquet that it coincides with the outer linear measure for any connected metric space. As corollaries we obtain simple proofs of Go{\l}\k{a}b's theorem (1928) on the lower semicontinuity of linear measure of continua and a theorem of Bogn\'ar (1989) on the linear measure of the closure of a set. We do not use any measure theory apart from the definition of outer linear measure.

Given any n points in the plane, not all on the same line, there exist two non-collinear triples such that the ratio of the areas of the triangles they determine, differs from 1 by at most $$O(\log n/n^2)$$ O(logn/n2) . If we furthermore insist that the two triangles have a common edge, then there are two with area ratios differing from 1 by at most O (1/ n ). This improves some results of Ophir and Pinchasi (Discrete Appl. Math. 174 (2014), 122–127). We also give some constructions for these and related problems.

An ordinary hypersphere of a set of points in real d-space, where no d+1 points lie on a (d-2)-sphere or a (d-2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d+1 points of the set. Similarly, a (d+2)-point hypersphere of such a set is one that contains exactly d+2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of (d+2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ⩾ 4.

Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\in S$ a distance $r(x)>0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erdős and Pach (1988) introduced the extremal quantity $f_3(n)=\max\sum_{x\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of $3$-space and all assignments $r\colon S\to(0,\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\mathcal{C}$ and the axis of symmetry $\mathcal{L}$ of $\mathcal{C}$, and $r(x)$ equals the distance from $x$ to $C$ for each $x\in S\cap\mathcal{L}$. This, together with a new construction, implies that $f_3(n)=n^2/4 + 5n/2 + O(1)$.

Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms.

Let x and y be two unit vectors in a normed plane R2 . We say that x is Birkhoff orthogonal to y if the line through x in the direction y supports the unit disc. A B-measure (Fankhänel in Beitr Algebra Geom 52(2):335-342, 2011) is an angular measure μ on the unit circle for which μ(C)=π/2 whenever C is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.

The dimension of a graph G is the smallest d for which its vertices can be embedded in d-dimensional Euclidean space in the sense that the distances between endpoints of edges equal 1 (but there may be other unit distances). Answering a question of Erdős and Simonovits (1980) [5], we show that any graph with less than (d+22) edges has dimension at most d. Improving their result, we prove that the dimension of a graph with maximum degree d is at most d. We show the following Ramsey result: if each edge of the complete graph on 2d vertices is coloured red or blue, then either the red graph or the blue graph can be embedded in Euclidean d-space. We also derive analogous results for embeddings of graphs into the (d−1)-dimensional sphere of radius 1/2.

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 in d -space. It is known that f(2)=7 , f(3)=10 , and that the extremal almost-equidistant sets are unique. We give independent, computer-assisted proofs of these statements. It is also known that f(5) ≥ 16 . We further show that 12≤ f(4)≤ 13 , f(5)≤ 20 , 18≤ f(6)≤ 26 , 20≤ f(7)≤ 34 , and f(9)≥ f(8)≥ 24 . Up to dimension 7, our work is based on various computer searches, and in dimensions 6–9, we give constructions based on the known construction for d=5 . For every dimension d ≥ 3 , we give an example of an almost-equidistant set of 2d+4 points in the d -space and we prove the asymptotic upper bound f(d) ≤ O(d^3/2) .

An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on Green and Tao's work on ordinary lines in the plane, combined with classical results on space quartic curves and non-generic projections of curves. This gives an alternative approach to Ball's recent results on ordinary planes, as well as extending them. We also give bounds on the number of coplanar quadruples determined by a finite set of points on a rational space quartic curve in complex 3-space, answering a question of Raz, Sharir, and De Zeeuw [Israel J. Math. 227 (2018) 663-690].

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least [Formula: see text] ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most [Formula: see text] circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most [Formula: see text] ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

We show that any k-th closed sphere-of-influence graph in a d-dimensional normed space has a vertex of degree less than $$5^d k$$ , thus obtaining a common generalization of results of Füredi and Loeb (Proc Am Math Soc 121(4):1063–1073, 1994 [1]) and Guibas et al. (Sphere-of-influence graphs in higher dimensions, Intuitive geometry [Szeged, 1991], 1994, pp. 131–137 [2]).

We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces.