Forcing positive partition relations

Abstract

We show how to force two strong positive partition relations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and use them in considering several well-known open problems.