Conservation of Ramsey’s theorem for pairs and well-foundedness Article Swipe

In this article, we prove that Ramsey’s theorem for pairs and two colors is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Pi ^1_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-conservative over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper R sans-serif upper C sans-serif upper A Subscript 0 Baseline plus sans-serif upper B normal upper Sigma 2 Superscript 0 plus sans-serif upper W sans-serif upper F left-parenthesis epsilon 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">B</mml:mi> </mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mi mathvariant="sans-serif">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>ϵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {RCA}_0 + \mathsf {B}\Sigma ^0_2 + \mathsf {WF}(\epsilon _0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper R sans-serif upper C sans-serif upper A Subscript 0 Baseline plus sans-serif upper B normal upper Sigma 2 Superscript 0 plus union Underscript n Endscripts sans-serif upper W sans-serif upper F left-parenthesis omega Subscript n Superscript omega Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">B</mml:mi> </mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:munder> <mml:mo>⋃</mml:mo> <mml:mi>n</mml:mi> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mi mathvariant="sans-serif">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> <mml:mi>ω</mml:mi> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {RCA}_0 + \mathsf {B}\Sigma ^0_2 + \bigcup _n \mathsf {WF}(\omega ^\omega _n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These results improve theorems from Chong, Slaman and Yang [Adv. Math. 308 (2017), pp. 121–141] and Kołodziejczyk and Yokoyama [<italic>In search of the first-order part of Ramsey’s theorem for pairs</italic>, Springer, Cham, 2021] and belong to a long line of research towards the characterization of the first-order part of Ramsey’s theorem for pairs.