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 RH(K1, ... ,Kr,n,r): For any r-colouring of En 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 Ki 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 E2 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 E2 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 E2, 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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