In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation ∆ +Moreover, we show that ∆ + R is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications.2010 Mathematics Subject Classification. Primary 03E15, Secondary 37B05.

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.

In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers n ≥ 3,where R cl (•, •) and R(•, •) denote the closed Ramsey numbers and the classical Ramsey numbers, respectively. We also establish the following asymptotically weaker upper bound:eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.

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