JAF 2024 Passau
Date:
The search for the first-order part of Ramsey’s theorem for pairs and two colors (\(\mathsf{RT}^2_2\)) has been a long-standing question in the field of reverse mathematics. \(\mathsf{RT}^2_2\) is known to be provable using \(\Sigma^0_2\) induction but is strictly weaker than it. It also implies a statement called \(\Sigma^0_2\)-collection (\(\mathsf{B}\Sigma^0_2\)). However, whether \(\mathsf{RT}^2_2\) is conservative over the base theory \(\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2\) for arithmetic statements remains an open question. In their 2016 article, Ludovic Patey and Keita Yokoyama showed that \(\mathsf{RT}^2_2\) was \(\Pi^0_3\) conservative over the base theory \(\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2\). In this talk, we provide a sketch of an improvement of that result using an elaboration of the method of indicators, by showing that \(\mathsf{RT}^2_2\) is even \(\Pi^0_4\) conservative over \(\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2\).