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Note on canonical partitions. (English) Zbl 0584.05006
Let \(A=(0,1,2,...\}\) and let r be any positive integer. A function f with domain \([A]^ r\) (the set of all r-element subsets of A) provides a partition of \([A]^ r\). For any subset L of \(\{\) 0,1,...,r-1\(\}\) and any subset B of A, the partition f is said to be L-canonical on B provided that for all \(\{x_ 0,x_ 1,...,x_{r-1}\}_ <,\{y_ 0,...,y_{r- 1}\}_ <\subseteq B\) we have \(f(\{x_ 0,...,x_{r-1}\})=f(\{y_ 0,...,y_{r-1}\})\) if and only if \(x_{\lambda}=y_{\lambda}\) for all \(\lambda\) in L. This note provides a relatively simple proof of the following theorem of P. Erdős and the author [J. Lond. Math. Soc. 25, 249-255 (1950; Zbl 0038.153)]: Given any partition f of \([A]^ r\), there is an infinite subset B of A and a set \(L\subseteq \{0,...,r-1\}\) such that f is L-canonical on B.
Reviewer: N.H.Williams

05A17 Combinatorial aspects of partitions of integers
05A05 Permutations, words, matrices
03E05 Other combinatorial set theory
subsets; partition
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