Borel-Tukey morphisms between Ramsey homogeneity relations

Tatsuya Goto

paper

Borel-Tukey morphisms between Ramsey homogeneity relations

arXiv:2609.23494

Abstract

We study Borel-Tukey morphisms between almost-homogeneity relations. For finite $r, s \geq 1$ and $m, n \geq 2$ , we give the complete one-stage classification: $Hom_{m}^{r} \to_{BT} Hom_{n}^{s}$ holds exactly when $r > s$ , or when $r = s$ and $m \geq n$ . Our main result is that binary triple homogeneity requires exactly three sequential binary pair stages: $Hom_{2}^{2}; Hom_{2}^{2}; Hom_{2}^{2} ⟶_{BT} Hom_{2}^{3}$ and $Hom_{2}^{2}; Hom_{2}^{2} \neq ⟶_{BT} Hom_{2}^{3}$ . We also construct Borel morphisms $Hom_{2}^{2}; Hom_{3}^{2} ⟶_{BT} Hom_{2}^{3}$ and $Hom_{2}^{2}; Hom_{2}^{2} ⟶_{BT} Hom_{k}^{2}$ for every finite positive $k$ .

36 pages