对《库罗夫卡笔记》中第17.102问题的否定回答

中文摘要 AI 辅助

本文研究《库罗夫卡笔记》中的第17.102问题并给出否定答案

英文摘要

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 $XAX\cap B=\varnothing$. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if $A$ and $B$ are disjoint and $|A|,|B|<|G|$, 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 $W_G(A,B)=\{g\in G:\{g,g^{-1}\}A\{g,g^{-1}\}\cap B=\varnothing\}$, 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 $Γ_{A,B}$, 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 $|A|$, $|B|$ and structural invariants of the group, independently of the specific position of $A$ and $B$.

发表机构

• School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）

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