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Gaps in

Authors:	Zoran Spasojevic

Institution:	Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Abstract:	For a partial order $(P,\le _P)$ , let $\Gamma (P,\le _P)$ denote the statement that for every $\le _P$ -increasing $\omega _1$ -sequence $a\subseteq P$ there is a $\le _P$ -decreasing $\omega _1$ -sequence $b\subseteq P$ on top of such that is an $(\omega _1,\omega _1)$ -gap in . The main result of this paper is that $\mathfrak t>\omega _1\leftrightarrow \Gamma(\mathcal P(\omega ),\subset ^)\leftrightarrow \Gamma (\omega ^\omega ,\le ^)$ . It is also shown, as a corollary, that $\Gamma (\omega ^\omega ,\le ^)\to \mathfrak b>\omega _1$ but $\mathfrak b>\omega _1\not \to\Gamma (\omega ^\omega ,\le ^)$ .

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