单变量单词的Carlson-Simpson引理的$Π^0_4$保守性
$Π^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words
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中文总结 AI 辅助
本文证明单变量单词的Carlson-Simpson引理的2色版本是二阶算术子系统的$\forall Π^0_4$保守扩张，解答了相关归纳强度问题。
中文摘要 AI 辅助
Carlson和Simpson证明，对有限字母表$A$上的每个有限着色的单变量单词，都存在一个无限ω变量单词，使得其上所有单变量单词都是单色的。该结论对应ℓ着色，记为$\textsf{CSL}^1_\textsf{ℓ}$，已知其严格弱于$\textsf{ACA}_0$。我们证明$\textsf{RCA}_0 + \textsf{CSL}^1_2$是$\textsf{RCA}_0 + \textsf{B}\textsf{Sigma}_2$的$\forall Π^0_4$保守扩张。其推论包括：通用无三角Henson图对2着色的不可分性、两色对的树定理，均不蕴含$\textsf{Σ}^0_2$归纳，这解答了Chong、Li、Wang和Yang提出的问题。
英文摘要
Carlson and Simpson proved that for every finite coloring of the 1-variable words over a finite alphabet~$A$, there is an infinite $ω$-variable word on which all the 1-variable words are monochromatic. This statement for $\ell$-colorings, written $\mathsf{CSL}^1_\ell$, is known to be strictly weaker than $\mathsf{ACA}_0$. We prove that $\mathsf{RCA}_0 + \mathsf{CSL}^1_2$ is a $\forall Π^0_4$-conservative extension of $\mathsf{RCA}_0 + \mathsf{B}Sigma_2$. Among its consequences, it implies that neither the indivisibility of the universal triangle-free Henson graph for 2-colorings, nor the tree theorem for pairs and two colors, imply $Σ^0_2$-induction. This answers a question of Chong, Li, Wang and Yang.
