Abstract
We present an abstract framework for canonizing partition theorems. The concept of attribute functions and of diversification allows us to establish a canonizing product theorem, generalizing previous results of R. Rado [ J. London Math. Soc. 29 (1954) , 71–83] for the situation of Ramsey's theorem. As applications we mention a canonizing product theorem for arithmetic progressions and for finite geometric arguesian lattices. We show that finite sets and finite vector spaces have the diversification property. Along these lines, iterated versions of the Erdös-Rado canonization theorem [ J. London Math. Soc. 25 (1950) , 249–255] and its q-analogue for finite vector spaces [ B. Voigt, Combinatorica 4 (1984) , 219–239] are derived.
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