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Regressive partition relations for infinite cardinals. (English) Zbl 0627.03031

Let X be a set of ordinals and n a natural number. A function f with domain \([X]^ n\) is said to be regressive if \(f(a)<\min a\) for all a in \([X]^ n\). The subset H of X is said to be min-homogeneous for f if \(f(a)=f(b)\) for all a, b in \([X]^ n\) with min a\(=\min b\). The symbol \(X\to (\gamma)^ n_{reg}\) means that whenever f with domain \([X]^ n\) is regressive, then there is H in \([X]^{\gamma}\) which is min- homogeneous for f. An earlier result of J. E. Baumgartner [J. Symb. Logic. 40, 541-554 (1975; Zbl 0325.04006)] gives that \((2^{\kappa})^+\to (\kappa^++1)^ 2_{reg}\). However, to achieve positive partition relations when the exponent n exceeds 2 requires large cardinals. One of the main results in this paper is that the following are equivalent: (a) \(\kappa\) is \((n+1)\)-Mahlo; (b) For any \(\gamma <\kappa\) and any unbounded \(X\subseteq \kappa\), \(X\to (\gamma)^{n+3}_{reg}\); (c) For any closed unbounded \(C\subseteq \kappa\), \(C\to (\omega)^{n+3}_{reg}\). (Here \(\kappa\) is 0-Mahlo iff \(\kappa\) is inaccessible, and \(\kappa\) is \((n+1)\)-Mahlo iff every closed unbounded subset of \(\kappa\) contains an n-Mahlo cardinal.) Related work on finite min-homogeneous sets appears in J. H. Schmerl [Trans. Am. Math. Soc. 188, 281-291 (1974; Zbl 0275.02060)].
Reviewer: N.H.Williams

MSC:

03E05 Other combinatorial set theory
03E55 Large cardinals
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References:

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