Logic Seminar

Please note that the time is PST

Let ΓΓ be a compact Polish group of finite Lebesgue covering dimension. For a countably infinite subset S⊆ΓS⊆Γ, a domatic ℵ0ℵ0-partition (for its Schreier graph on ΓΓ) is a partial function f:Γ⇀Nf:Γ⇀N such that for every x∈Γx∈Γ, one has f[S⋅x]=Nf[S⋅x]=N. We show that a continuous domatic ℵ0ℵ0-partition exists, if and only if a Baire measurable domatic ℵ0ℵ0-partition exists, if and only if the topological closure of SS is uncountable. A Haar measurable domatic ℵ0ℵ0-partition exists for all choices of SS.

We will talk about an application of this result to the theory of sum sets in RnRn, and if time permits, some other examples of domatic partitions in the descriptive graph combinatorial setting. This work is based on an undergraduate thesis project under the supervision by Clinton Conley.