Recent trends in Euclidean Ramsey theory. (English) Zbl 0816.05060
The paper surveys several new results in Euclidean Ramsey theory. A finite set \(X\) in some Euclidean space \(E^k\) is called Ramsey if for all \(r\) there exists some \(N= N(X, r)\) such that for every partition of \(E^N\) into \(r\) subsets, some of these subsets contain a congruent copy of \(X\). Basic results concerning the unsolved question on the characterization of Ramsey sets are discussed. Also the question on the restriction to spheres is addressed. Moreover, density theorems, partition variants, results on the chromatic number of \(E^n\) (two points are joined by an edge if their distance is 1) and partition theorems in fixed dimension together with difficult conjectures are presented.
Reviewer: Konrad Engel (Rostock)
Keywords:
Euclidean Ramsey theory; Euclidean space; partition; congruent copy; Ramsey sets; spheres; density theorems; chromatic numberReferences:
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