On small analytic relations
Streszczenie
We study the class of analytic binary relations on Polish spaces, compared by the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is ${\bf \Sigma }^{0}_{\xi }$ (or ${\bf \Pi }^{0}_{\xi }$), for $\xi \geq 3$, 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 up of locally countable Borel relations minimal among non-${\bf \Sigma }^{0}_2$ (or non-${\bf \Pi }^{0}_2$) 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 not necessarily locally countable case, under 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).
