On small analytic relations

TL;DR

This paper characterizes analytic binary relations on Polish spaces using antichain bases, providing concrete classifications for different complexity levels and extending results in topological Ramsey theory.

Contribution

It introduces finite antichain bases for classes of analytic relations based on their descriptive set-theoretic complexity, with new results for locally countable and uncountable relations.

Findings

Finite antichain basis for locally countable Borel relations when ξ ≥ 3

Concrete antichain of size continuum for relations at ξ=2

Positive results for acyclic relations and uncountable analytic relations

Abstract

We study the class of analytic binary relations on Polish spaces, compared with the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is $\Sigma$ 0 $\xi$ (or $\Pi$ 0 $\xi$), when $\xi$ $\ge$ 3, by providing a concrete finite antichain basis. We give a similar characterization for arbitrary relations when $\xi$ = 1. When $\xi$ = 2, we provide a concrete antichain of size continuum made of locally countable Borel relations minimal among non-$\Sigma$ 0 2 (or non-$\Pi$ 0 2) relations. The proof of this last result allows us to strengthen a result due to Baumgartner in topological Ramsey theory on the space of rational numbers. We prove that positive results hold when $\xi$ = 2 in the acyclic case. We give a general positive result in the non-necessarily locally countable case, with another suitable acyclicity assumption. We provide a concrete finite antichain basis for the class of uncountable analytic relations. Finally, we deduce from our positive results some antichain basis for graphs, of small cardinality (most of the time 1 or 2).