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Rainbow Ramsey theorems for colorings establishing negative partition relations. (English) Zbl 1142.03027

Summary: Given a function \(f\), a subset of its domain is a rainbow subset for \(f\) if \(f\) is one-to-one on it. We start with an old Erdős problem: Assume \(f\) is a coloring of the pairs of \(\omega_1\) with three colors such that every subset \(A\) of \(\omega_1\) of size \(\omega_1\) contains a pair of each color. Does there exist a rainbow triangle? We investigate rainbow problems and results of this style for colorings of pairs establishing negative “square bracket” relations.

MSC:

03E02 Partition relations
03E05 Other combinatorial set theory
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