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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 313.05002

**Autor: ** Erdös, Paul; Graham, Ronald L.; Montgomery, P.; Rothschild, B.L.; Spencer, Joel; Straus, E.G.

**Title: ** Euclidean Ramsey theorems. II, III. (In English)

**Source: ** Infinite finite Sets, Colloq. Honour Paul Erdös, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 529-558, 559-583 (1975).

**Review: ** [For the entire collection see Zbl 293.00009.]

Continuing the investigations begun in Part I [J. combinat. Theory, Ser. A 14, 341-363 (1973; Zbl 276.05001)], the authors consider several generalizations. In Section 2 of Part II the statement R(K,n,r) defined in Part I (vide Zbl review) is generalized to R_{H}(K_{1}, ... ,K_{r},n,r): For any r-colouring of E^{n} there is some i and some K'_{i} consisting only of points of the i-th color, such that K'_{i} is the image of K_{i} under some element of H.'' Several explicit results are further generalized in the following section, where the question of the existence of many copies of a particular configuration is considered. Section 4 is mainly concerned with the existence of infinite configurations in finite- and infinite-dimensional Euclidean spaces and in real Hilbert spaces. The final section of Part II considers colourings of the edges of Euclidean spaces – i.e. of the unordered point-pairs. For example, it is proved that ``Theorem 24. The edges of E^{2} can be line colored with 2 colors so that no triangle with all angles at most 90^{\circ} has all three edges the same color. On the other hand, for every line coloring of E^{2} with 2 colors and every \epsilon > 0 some triangle with all angles less than 90^{\circ}+\epsilon has all three edges the same color. For very \epsilon > 0 and every 2-coloring of the edges of E^{2}, some triangle with all angles at most 180^{\circ}+\epsilon has all three edges the same color.'' Part III is devoted to many variations of the special case of R(**{**a,b,c **}**,2,2), and related problems.

**Reviewer: ** W.G.Brown

**Classif.: ** * 05A05 Combinatorial choice problems

05C15 Chromatic theory of graphs and maps

04A20 Combinatorial set theory

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