On small analytic relations

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).