Ultrafilters and decompositions of Abelian groups (Q2754825)

It follows from the well-known theorem of Ramsey that for any \(2\)-coloring of an infinite Abelian group there exists such an infinite subset \(A\subseteq G\) that the subset \(PS(A)\) is monochromatic (here \(PS(A)=\{a+b:a, b\in A, a\not= b\}\)). A free ultrafilter \(\phi\) on a group \(G\) is said to be a \(PS\)-ultrafilter if there exists such a subset \(A\in \phi .\) All Ramsey ultrafilters are \(PS\)-ultrafilters but the converse statement is false as the author earlier proved (see \textit{I. V. Protasov} [Mat. Stud. 7, No. 2, 133-138 (1997; Zbl 0927.22006)]). It is proved that every \(PS\)-ultrafilter on a group without \(2\)-torsion is a Ramsey ultrafilter, moreover the study of non-Ramsey \(PS\)-ultrafilters was reduced to the case of \(2\)-groups. The paper contains also many results concerning the subselectivity of free ultrafilters. The main tool of investigation is the semigroup of ultrafilters on an Abelian group.