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Gaps in
Authors:Zoran Spasojevic
Affiliation: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 $asubseteq P$ there is a $le _P$-decreasing $omega _1$-sequence $bsubseteq P$ on top of $a$ such that $(a,b)$ is an $(omega _1,omega _1)$-gap in $P$. The main result of this paper is that $mathfrak t>omega _1leftrightarrow Gamma(mathcal P(omega ),subset ^*)leftrightarrow Gamma (omega ^omega ,le ^*)$. It is also shown, as a corollary, that $Gamma (omega ^omega ,le ^*)to mathfrak b>omega _1$ but $mathfrak b>omega _1not toGamma (omega ^omega ,le ^*)$.

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