Kourovka Notebook Problem 15.78

Definitions and mathematical statement

Let $n$ be a positive integer, let $F$ be a free group of any rank, and let $c:F\to\{1,\ldots,n\}$ be an $n$-colouring. A subset $S\subseteq F$ is monochromatic if there is an $i\in\{1,\ldots,n\}$ such that $c(s)=i$ for every $s\in S$.

The source does not define symmetry with respect to a group element. Under the usual point-reflection interpretation, symmetry with respect to $g\in F$ means invariance under the involution $s\mapsto gs^{-1}g$, or equivalently \[ S=gS^{-1}g,\qquad S^{-1}=\{s^{-1}:s\in S\}. \] Under this interpretation, and without adding a size requirement absent from the record, the supplied question has the form \[ \begin{gathered} \text{For every positive integer }n,\text{ every free group }F\text{ of any rank,}\\ \text{and every }c:F\to\{1,\ldots,n\},\text{ do there exist }g\in F,\ S\subseteq F,\ i\in\{1,\ldots,n\}\\ \text{such that }S=gS^{-1}g\quad\text{and}\quad (\forall s\in S)\ c(s)=i\ ? \end{gathered} \] The source supplies no requirement that $S$ be nonempty, infinite, or of any prescribed cardinality. Its intended size condition therefore remains unresolved; the display records only the literal size-unrestricted reading under the stated symmetry convention. In particular, an infinitude requirement must not be silently inserted. If such a requirement was intended, the permitted ranks would also need clarification. The source states that the assertion is true for $n=2$, but the precise size and symmetry conventions governing that assertion are not supplied.

Short English statement

The question asks whether every finite colouring of a free group of any rank has a monochromatic subset symmetric about some group element. The source states a positive result for two colours. The meaning of symmetry and the required size of the subset need clarification before the intended problem can be stated precisely.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $701166$, exact source problem code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied reference: The Kourovka Notebook (Unsolved Problems in Group Theory), problem 15.78, 15th issue (2002), retained in the supplied 21st-issue edition (2026), arXiv:1401.0300v48. The collection metadata names E. I. Khukhro and V. D. Mazurov as editors. No external reference was independently consulted.
• Attribution: the background and proposer field name I. V. Protasov; the statement itself begins with the parenthetical attribution R. I. Grigorchuk. Their respective roles are not explained in the supplied record.
• The registry labels the problem open and supplies the assertion that the two-colour case is true. These are supplied source claims, not independently verified mathematical or literature assessments.