2017/09/13 by William Chan, Chan, William, Connor Meehan +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1709.04567
openalex publication_date 2017/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If X is a set, E is an equivalence relation on X, and n ∈ ω, then define [X]nE = \(x0, ..., xn - 1) ∈ nX : (∀ i,j)(i ≠ j ⇒ ¬(xi E xj))\. For n ∈ ω, a set X has the n-Jónsson property if and only if for every function f : [X]n_= → X, there exists some Y ⊆ X with X and Y in bijection so that f[[Y]n_=] ≠ X. A set X has the Jónsson property if and only for every function f : (\bigcupn ∈ ω[X]n_=) → X, there exists some Y ⊆ X with X and Y in bijection so that f[\bigcupn ∈ ω [Y]n_=] ≠ X. Let n ∈ ω, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C ⊆ nX, there is some Δ11 A ⊆ X so that E ≤Δ11 E \upharpoonright A and [A]nE ⊆ C. The following equivalence relations will be considered: E0 is defined on ω2 by x E0 y if and only if (∃ n)(∀ k > n)(x(k) = y(k)). E1 is defined on ω(ω2) by x E1 y if and only if (∃ n)(∀ k > n)(x(k) = y(k)). E2 is defined on ω2 by x E2 y if and only if ∑\(1)/(n + 1) : n ∈ x \triangle y\ < ∞, where \triangle denotes the symmetric difference. E3 is defined on ω(ω2) by x E3 y if and only if (∀ n)(x(n) E0 y(n)). Holshouser and Jackson have shown that ℝ is Jónsson under AD. It will be shown that E0 does not have the 3-Mycielski property and that E1, E2, and E3 do not have the 2-Mycielski property. Under ZF + AD, ω2 / E0 does not have the 3-Jónsson property.