Problem #8

Conservativity of Ramsey's Theorem for pairs

Open!!

Is $\mathsf{RT}^2_2$ , i.e. Ramsey's Theorem for pairs and two colors, $\Pi^1_1$ -conservative over $\mathsf{RCA}_0 + B\Sigma^0_2$ ?

Reference for the problem statement

C. T. Chong, Theodore A. Slaman and Yue Yang, The inductive strength of Ramsey's Theorem for Pairs, Advances in Mathematics, 2017 [link] [doi]

Known Partial Results

Hirst showed that $\mathsf{RT}^2_2$ proves $B\Sigma^0_2$ .
Chong, Slaman and Yang showed that $\mathsf{RT}^2_2$ does not prove $I\Sigma^0_2$ over $\mathsf{RCA}_0$ .
Patey and Yokoyama showed that $\mathsf{RT}^2_2$ is $\Pi^0_3$ -conservative over $\mathsf{RCA}_0$ .
Cholak, Jockusch and Slaman showed that $\mathsf{RT}^2_2 + I\Sigma^0_2$ is $\Pi^1_1$ -conservative over $\mathsf{RCA}_0 + I\Sigma^0_2$ .

This is one of the most well-known open problems in reverse math.

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