Title: Higman’s Lemma is Stronger for Better Quasi Orders
Abstract: Abstract We prove that Higman’s lemma is strictly stronger for better quasi orders than for well quasi orders, within the framework of reverse mathematics. In fact, we show a stronger result: the infinite Ramsey theorem (for tuples of all lengths) is equivalent to the statement that any array $$[\mathbb N]^{n+1}\rightarrow \mathbb N^n\times X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>N</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> for a well order X and $$n\in \mathbb N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> is good, over the base theory $$\mathsf {RCA_0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>RCA</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> .