$Π^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words

Quentin Le Houérou; Ludovic Patey

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arXiv · 2607.28116

$Π^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words

Quentin Le Houérou·Ludovic Patey

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Abstract

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.

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BibTeX RIS

Quentin Le Houérou, Ludovic Patey. 2026-07-30. $Π^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words. https://arxiv.org/abs/2607.28116

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