Metric Spaces in Which Many Triangles Are Degenerate

Autor: Chvátal, Vašek, de Rancourt, Noé, Quintero, Guillermo Gamboa, Kantor, Ida, Szabó, Péter G. N.

Richmond and Richmond (American Mathematical Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove

Externí odkaz: http://arxiv.org/abs/2401.05259

Sparse metric hypergraphs

Autor: Chvátal, Vašek, Quintero., Guillermo A. Gamboa, Kantor, Ida

Given a metric space $(X, \rho)$, we say $y$ is between $x$ and $z$ if $\rho(x,z) = \rho(x,y) + \rho(y,z)$. A metric space gives rise to a 3-uniform hypergraph that has as hyperedges those triples $\{ x,y,z \}$ where $y$ is between $x$ and $z$. Such

Externí odkaz: http://arxiv.org/abs/2310.16993

Metric hypergraphs and metric-line equivalences

Autor: Chvátal, Vašek, Kantor, Ida

In a metric space $M=(X,d)$, we say that $v$ is between $u$ and $w$ if $d(u,w)=d(u,v)+d(v,w)$. Taking all triples $\{u,v,w\}$ such that $v$ is between $u$ and $w$, one can associate a 3-uniform hypergraph with each finite metric space $M$. An effort

Externí odkaz: http://arxiv.org/abs/2207.11811

A De Bruijn-Erd\H{o}s theorem in graphs?

Autor: Chvátal, Vašek

Publikováno v: In: Graph Theory Favorite Conjectures and Open Problems - 2 (Ralucca Gera, Teresa W. Haynes, and Stephen T. Hedetniemi, eds.), Springer (2018), pp. 149--176

A set of $n$ points in the Euclidean plane determines at least $n$ distinct lines unless these $n$ points are collinear. In 2006, Chen and Chv\'atal asked whether the same statement holds true in general metric spaces, where the line determined by po

Externí odkaz: http://arxiv.org/abs/1812.06288

McCulloch-Pitts brains and pseudorandom functions

Autor: Chvátal, Vašek, Goldsmith, Mark, Yang, Nan

Publikováno v: Neural Computation 28 (2016), 1042--1050

In a pioneering classic, Warren McCulloch and Walter Pitts proposed a model of the central nervous system. Motivated by EEG recordings of normal brain activity, Chv\'atal and Goldsmith asked whether or not these dynamical systems can be engineered to

Externí odkaz: http://arxiv.org/abs/1603.01573