| Abstract: |
In this talk, we will revisit the effective techniques used in the original proofs of the G0 dichotomy
(Kechris-Solecki-Todorcevic) and the E0 dichotomy (Harrington-Kechris-Louveau),
and use them to give a new proof of the (G,0, H0) dichotomy (Miller). The (G0, H0)
dichotomy states that if E is an analytic equivalence relation and G is an analytic graph, both on a Polish space X,
then exactly one of the following occurs: EITHER
- there is a smooth equivalence relation F containing E and a Borel ω-coloring of G that is
proper whenever restricted to a single-F class OR
- there is a continuous homomorphism of (G0, H0) to (G, E),
where G0 and H0 are both some fixed graphs on Cantor space.
Miller's approach was to assume the negation of (2) and use the failure to find an
embedding in order to construct F and the desired coloring.
Instead, we will identify a set of purely topological conditions that are sufficient to find a
homomorphism of (G0, H0) to (G, E) and, assuming the negation of (1), we will use the Gandy-Harrington topology
to find a Polish subspace exhibiting these properties.
In return for using some effective techniques, we will get an arguably simpler splitting construction.
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