Old and new problems and results in combinatorial number theory

The author discusses various solved and unsolved geometrical problems all of which are of a combinatorial nature. Some are of metrical character and some are more number theoretic.

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On some problems of elementary and combinatorial geometry

In a series of papers, Erdos et al. [E] have investigated this property. They have shown that all Ramsey sets are spherical, that is, every Ramsey set is contained in an appropriate sphere. On the other hand, they have shown that the vertex set (and, therefore, all its subsets) of bricks (d-dimensional parallelepipeds) is Ramsey. The simplest sets that are spherical but cannot be embedded into the vertex set of a brick are the sets of obtuse triangles. In [FR1], it is shown that they are indeed Ramsey, using Ramsey's Theorem (cf. [G2]) and the Product Theorem of [E]. The aim of the present paper is twofold. First, we want to show that the vertex set of every nondegenerate simplex in any dimension is Ramsey. Second, we want to show that for both simplices and bricks, and even for their products, one can in fact choose n(r, B) = c(B)logr, where c(B) is an appropriate positive constant. The paper is organized as follows. In ?2, super-Ramsey property is introduced. This notion is stronger than being Ramsey. It is shown that the direct product of super-Ramsey sets is super-Ramsey. In ?3, it is shown that if every edge of an n-dimensional simplex is between 1 e and 1 + e with e = e(n) being a sufficiently small positive number, then it can be embedded into a brick, i.e., into the direct product of two-element sets. In ?4, it is proved that given a nondegenerate simplex with edge lengths aij, 1 0, there exists some super-Ramsey simplex whose edge lengths vi verify la 2_-v2jI <d5 for all 1 < < i < n. Intheproofofthisresult, an earlier result

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A partition property of simplices in Euclidean space

On the space chromatic number

In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore the different aspects of this connection.

Structural Ramsey theory of metric spaces and topological dynamics of isometry groups

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Peter Montgomery

Papers

Euclidean Ramsey Theorems. I