Equilateral Triangles in Finite Metric Spaces

On Metric Ramsey-type Dichotomies

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form ”Every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral”. We consider two distinct interpretations fo...

ar X iv : 1 60 3 . 01 90 7 v 1 [ m at h . C A ] 7 M ar 2 01 6 Equilateral triangles in subsets of R d of large Hausdorff dimension

We prove that subsets of R, d ≥ 4 of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two dimensions (see [3]). We show that no extra conditions are needed in dimensions four and higher. The three dimensional case remains open. Some interesting parallels exist betw...

2 1 A ug 2 00 7 The geometry of Minkowski spaces — a survey . Part I Horst Martini

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonici...

Metric Geometry and Random Discrete Morse Theory

How to put a metric on a given simplicial complex? One way is to declare all edges to have unit length, and to regard all triangles as equilateral triangles in the Euclidean plane. This yields the equilateral flat metric, also known as regular metric. Many other options are possible; for example, one can assign different lengths to the various edges. The metric is called acute (resp. non-obtuse...

Equilateral Triangles in Finite