Separability of Subsets in Infinite Groups

We study the separability of subsets of an infinite group. Given an infinite group $G$ and subsets $A,B\subset G$, we say that $A$ and $B$ are separated in $G$ if there exists an infinite symmetric subset $X\subset G$ with $e\in X$ and $XAX\cap B=\mathrm{\varnothing }$. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if $A$ and $B$ are disjoint and $\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }<\mathrm{\mid }G\mathrm{\mid }$, must they be separated? We first give a negative answer, constructing two essentially different families of counterexamples, based respectively on rigid binary relations and on torsion-free groups whose square sets have cardinality smaller than that of the group. On this basis we introduce the $W$-witness set $W_G(A,B)=\{g\in G:\{g,g^{-1}\}A\{g,g^{-1}\}\cap B=\mathrm{\varnothing }\}$, obtain a necessary and a sufficient condition for separability given by its size, and study the critical range. The central result of this paper is a necessary and sufficient characterization of separability: taking the inverse pairs $\{g,g^{-1}\}$ as vertices, we construct the conflict graph ${\mathrm{\Gamma }}_{A,B}$, and $A$ and $B$ are separated if and only if the associated graph contains an infinite independent set; separability is thereby converted into an ordinary graph-theoretic problem. From this we further derive the case of finite subsets, the case of Abelian groups, and several cardinal criteria, and we obtain, via Ramsey’s theorem, a structural dichotomy for the non-separable case. In addition, we study the range of cardinalities of counterexample groups and the possible sizes of $W$-witness sets, and prove that no uniform necessary and sufficient criterion can depend only on $\mathrm{\mid }A\mathrm{\mid }$, $\mathrm{\mid }B\mathrm{\mid }$ and structural invariants of the group, independently of the specific position of $A$ and $B$.