Dominating the Erdos-Moser theorem in reverse mathematics

Erdos-Moser and ISigma_2

The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erdős-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. Specifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISig…

The reverse mathematics of the Thin set and Erdős-Moser theorems

The thin set theorem for $n$-tuples and $k$ colors ($\mathsf{TS}^n_k$) states that every $k$-coloring of $[\mathbb{N}]^n$ admits an infinite set of integers $H$ such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\mathsf{TS}^n_k$, nor the free set theorem ($\mathsf{FS}^n$) imply the Erdős-Moser theorem ($\mathsf{EM}$) whenever $k$ is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $…

算术归约下 Erdős-Moser 定理的弱性

The weakness of the Erdős-Moser theorem under arithmetic reductions

本文研究 Erdős-Moser 定理在算术层谱下的计算复杂性，证明其 Δ⁰_n 实例具有 low_{n+1} 解，且非算术集合在相对化解中仍保持非算术性，并给出逐层精细化结果。

Degrees bounding principles and universal instances in reverse mathematics

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove that the stable version of the ascending descending sequence principle (SADS) as well as the stable version of the thin set theorem for pairs (STS(2)) do not admit a bound of low_2 degree. Therefore no principle between Ramsey's theorem for pairs RT22 and SADS or STS(2) admit a universal instance. We construct a low_2 degree bounding the Erdős-Moser theorem (EM…

有界Ramsey定理对二元对的逆数学研究

The reverse mathematics of bounded Ramsey's theorem for pairs

研究退化版Ramsey定理对二元对和两色（BRT²₂）的逆数学强度，证明其可由Erdős-Moser定理推出但不蕴含升降序列原理，且其可计算真版本等价于IΣ₂⁰。