Some new results on Borel irreducibility of equivalence relations

We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ERs of a family which includes Polish actions of $S_\infty$ (the group of all permutations of $\mathbb N$ and is closed under the Fubini product modulo the ideal Fin of all finite sets and under some other operations. We show that $\mathsf T_2$ (an equivalence relation called the equality of countable sets of reals is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with $\mathbf G_{\delta\sigma}$ classes and some ideals, and is closed under the Fubini product modulo Fin. These results and their corollaries extend some earlier irreducibility theorems of Hjorth and Kechris.