arXiv subjects

Vašek Chvátal

Publications and source records attributed to Vašek Chvátal.

13 recordsLinked to original sources

arXiv2024

Metric Spaces in Which Many Triangles Are Degenerate

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 that the hypothesis is unnecessarily strong: In a metric space on $n$ points, fewer than $7n^2/6$ suitably placed degenerate triangles suffice. However, fewer than $n(n-1)/2$ degenerate triangles, no matter how cleverly placed, never suffice.

math.CO↗

arXiv2023

Metric spaces in which many triangles are degenerate

Richmond and Richmond (Amer. Math. 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 that the hypothesis is unnecessarily strong: In fact, $Θ(n^2)$ suitably placed degenerate triangles suffice.

math.MG↗

arXiv2023

Sparse metric hypergraphs

Given a metric space $(X, ρ)$, we say $y$ is between $x$ and $z$ if $ρ(x,z) = ρ(x,y) + ρ(y,z)$. 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 hypergraphs are called metric and understanding them is key to the study of metric spaces. In this paper, we prove that hypergraphs where small subsets of vertices induce few edges are metric. Additionally, we adapt the notion of sparsity with respect to monotone increasing functions, classify hypergraphs that exhibit this version of sparsity and prove that they are metric.

math.CO↗

arXiv2022

Metric hypergraphs and metric-line equivalences

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 to solve some basic open questions regarding finite metric spaces has motivated an endeavor to better understand these associated hypergraphs. In answer to a question posed in arXiv:1112.0376, we present an infinite family of hypergraphs that are non-metric, i.e., they don't arise from any metric space. Another basic structure associated with a metric space is a binary equivalence on the vertex set, where two pairs are in the same class if they induce the same line. An equivalence that comes from some metric space is a metric-line equivalence. We present an infinite family of so called obstacles, that is, binary equivalences that prevent an equivalence from being a metric-line equivalence.

math.CO↗

arXiv2018

A De Bruijn-Erdős theorem in graphs?

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 Chvátal asked whether the same statement holds true in general metric spaces, where the line determined by points $x$ and $y$ is defined as the set consisting of $x$, $y$, and all points $z$ such that one of the three points $x,y,z$ lies between the other two. The conjecture that it does hold true remains unresolved even in the special case where the metric space arises from a connected undirected graph with unit lengths assigned to edges. We trace its curriculum vitae and point out twenty-nine related open problems plus three additional conjectures.

math.CO↗

arXiv2016

Patterns of conjunctive forks

Three events in a probability space form a conjunctive fork if they satisfy specific constraints on conditional independence and covariances. Patterns of conjunctive forks within collections of events are characterized by means of systems of linear equations that have positive solutions. This characterization allows patterns of conjunctive forks to be recognized in polynomial time. Relations to previous work on causal betweenness and on patterns of conditional independence among random variables are discussed.

math.PR↗

arXiv2016

Brains and pseudorandom generators

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 this model can be engineered to provide pseudorandom number generators. We supply evidence suggesting that the answer is negative.

math.DS↗

arXiv2016

McCulloch-Pitts brains and pseudorandom functions

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, Chvátal and Goldsmith asked whether or not these dynamical systems can be engineered to produce trajectories which are irregular, disorderly, apparently unpredictable. We show that they cannot build weak pseudorandom functions.

math.DS↗

arXiv2013

Number of lines in hypergraphs

Chen and Chvátal introduced the notion of lines in hypergraphs; they proved that every 3-uniform hypergraph with $n$ vertices either has a line that consists of all $n$ vertices or else has at least $\log_2 n$ distinct lines. We improve this lower bound by a factor of $2-o(1)$.

math.CO↗

arXiv2012

Can brains generate random numbers?

Motivated by EEG recordings of normal brain activity, we construct arbitrarily large McCulloch-Pitts neural networks that, without any external input, make every subset of their neurons fire in some iteration (and therefore in infinitely many iterations).

math.DS↗

arXiv2011

Finite Sholander Trees, Trees, and their Betweenness

We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.

math.CO↗

arXiv2009

A de Bruijn - Erdős theorem and metric spaces

De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.

math.CO↗

arXiv2006

Problems related to a de Bruijn - Erdős theorem

De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on related extremal combinatorial problems.

math.CO↗