This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equ...

Let K be a locally compact hypergroup. In this paper we initiate the concept of fundamental domain in locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fun...

Let E ⊆ F and E′ ⊆ F ′ be Borel equivalence relations on the standard Borel spaces X and Y , respectively. The pair (E,F ) is simultaneously Borel reducible to the pair (E′, F ′) if there is a Borel function f : X → Y that is both a reduction from E to E′ and a reduction from F to F ′. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs E ⊆ F of smooth ...

We prove: Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a (κ+ 1)-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph G0 into G. Theorem 2. In the Solovay model , every OD graph G on reals satisfies one o...

Let S∞ be the group of all permutations of N and X be a standard Borel space. Then the space XN of functions from N to X is a standard Borel space, and S∞ acts on this space by permutation where given x ∈ XN and g ∈ S∞, g · x(n) = x(g−1(n)). Given any countable subgroup G of S∞, we can likewise restrict this action to G and consider the induced orbit equivalence relation on XN, which we will no...

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.