Related papers: A de Bruijn - Erd\H{o}s theorem and metric spaces

In this work we show that any connected locally connected graph defines a metric space having at least as many lines as vertices with only three exception: the complete multipartite graphs $K_{1,2,2}$, $K_{2,2,2}$ and $K_{2,2,2,2}$. This…

In a recent paper Kr\'olak and Beem have shown differentiability of Cauchy horizons at all points of multiplicity one. In this note we give a simpler proof of this result.

In this paper, we introduced two new generalized metric spaces called partial b_{v}(s) and b_{v}({\theta}) metric spaces which extend b_{v}(s) metric space, b-metric space, rectangular metric space, v-generalized metric space, partial…

We consider the nonlinear Schr\"odinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering…

Einstein equations for several matter sources in homogeneous, isotropic metric are shown to reduce to a second order nonlinear ordinary differential equation. An analysis of its solutions is made in an important case.

This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…

In [1] M. Baker and S. Norine developed a theory of divisors and linear systems on graphs, and proved a Riemann-Roch Theorem for these objects (conceived as integer-valued functions on the vertices). In [2] and [3] the authors generalized…

We study the observability of the one-dimensional Schr{\"o}dinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series,…

In this paper, using the compression method, we recover the lower bound for the Erd\H{o}s unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $\mathbb{R}^k$ for all $k\geq 2$, we…

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been…

Let $h\colon\mathbb{Z}^+\to\mathbb{Z}^+\setminus\{1\}$ be any function such that $h(n)$ and $\lceil n^{1/h(n)}\rceil$ are computable from $n$ in $O(h(n)\cdot n^{1+1/h(n)})$ time. We show that given any $n$-point metric space $(M,d)$, the…

We prove relations among the classes of certain divisors on the moduli spaces of curves with marked points, generalizing the Brill-Noether Ray Theorem of Eisenbud and Harris.

We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…

For variational problems with $O(N)$-symmetry the existence of several geometrically distinct solutions had been shown by use of group theoretic approach in previous articles. It was done by a crafty choice of a family $H_i \subset O(N)$…

We prove that $n$ plane algebraic curves determine $O(n^{(k+2)/(k+1)})$ points of $k$-th order tangency. This generalizes an earlier result of Ellenberg, Solymosi, and Zahl on the number of (first order) tangencies determined by $n$ plane…

Erd\H{o}s and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that…

Let $\mathcal{P}$ be a set of $n$ points in the Euclidean plane. We prove that, for any $\epsilon > 0$, either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors determined by…

Second order scalar ordinary differential equations ({\sc ode}s) which are linearizable possess special types of symmetries. These are the only symmetries which are non fiber-preserving in the linearized form of the equation, and they are…

We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and…

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…