Euclidean Ramsey Theorems. I
01 May 1973-Journal of Combinatorial Theory, Series A (Academic Press Inc.)-Vol. 14, Iss: 3, pp 341-363
TL;DR: Questions of whether or not certain R are r -Ramsey where B is a Euclidean space and R is defined geometrically are investigated.
About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-05-01 and is currently open access. It has received 115 citations till now. The article focuses on the topics: Euclidean space.
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TL;DR: In this article, 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.
Abstract: 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.
90 citations
TL;DR: In this article, it was shown that the vertex set of every non-degenerate simplex in any dimension is Ramsey, and that the direct product of super-Ramsey sets is also Ramsey.
Abstract: 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
68 citations
TL;DR: The chromatic number of the space is the minimum number of colors needed to color all points of the Euclidean space and it is shown that this number is at least 6, improving the best-known previous bound of 5.
62 citations
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TL;DR: In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces is closely related to the combinatorial behavior of the class of their finite metric spaces.
Abstract: 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.
55 citations
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TL;DR: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula.
Abstract: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula*. But in the course of this investigation it is necessary to use certain theorems on combinations which have an independent interest and are most conveniently set out by themselves beforehand.
2,223 citations
"Euclidean Ramsey Theorems. I" refers background in this paper
...For instance, suppose A is the set of Z-subsets of an n-set S, and B is the set of k-subsets ofS.LetR={(a,b)IbCa}. THEOREM 1 (Ramsey [ 7 ])....
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TL;DR: In this paper, it was shown that a set is n-regular in X if, for any partition of X into N parts, some part has as a subset a member of the set.
Abstract: 1.Introduction. Suppose X is a set, 𝒞 a collection of sets (usually subsets of X), and N is cardinal number. Following the terminology of Rado [1], we say 𝒞 is N-regular in X if,for any partition of X into N parts, some part has as a subset a member of 𝒞. if 𝒞 is n-regular in X for each integer n, we say 𝒞 is regular in X.
442 citations
244 citations
Book•
01 Jan 1964
206 citations
"Euclidean Ramsey Theorems. I" refers background in this paper
...Proof. We refer the reader to [4] and [ 2 ] for proofs....
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