The paper under review is about the Borel combinatorics of the shift graphs of marked groups. Problems in this area arise from studying free Borel actions of countable groups. \par A graph $G \subset X \times X$ on a Polish space $X$ is called Borel if and only if it is Borel as a subset of the product space. As usual a {\it $Y$-coloring} of a graph $G \subset X \times X$ is a map $c \colon X \to Y$ such that ${x\mathbin{G}y} \Rightarrow {c(x) \neq c(y)}$. A $k$-coloring $c \colon X \to \{0,\dotsc, k-1\}$ of $G$ is called {\it Borel} if and only if it is a Borel function. The {\it Borel chromatic number} of $G$, denoted by $\chi_{B}(G)$, is the smallest $k$ such that $G$ admits a Borel $k$-coloring. \par Any countable group $\Gamma$ acts on the Polish space $2^{\Gamma}$ by the {\it left shift action}, which is given by $(g \cdot x)(h) = x(g^{-1}\cdot h)$. In particular $\Gamma$ acts on the free part $$ F(2^{\Gamma}) = \{x\in 2^{\Gamma} \mid \forall g \in \Gamma \,(g\neq 1_{\Gamma} \Rightarrow g\cdot x \neq x) \} $$ regarded as a Polish subspace of $2^{\Gamma}$. A {\it marked group} is a pair $(\Gamma, S)$, where $\Gamma$ is a (usually infinite) finitely generated group and $S$ is a symmetric set of generators not containing $1_{\Gamma}$. Finally, the {\it shift graph} of $(\Gamma, S)$, also denoted by $F(2^{\Gamma})$, is the graph on $F(2^{\Gamma})$ given by putting an edge between $x$ and $y$ exactly when $g \cdot x = y$ for some $g \in S$. \par When $\Gamma$ is finite, $F(2^{\Gamma})$ is a disjoint union of finitely many isomorphic copies of ${\rm Cay}(\Gamma)$, the Cayley graph of $\Gamma$. Thus Borel combinatorial properties of $F(2^{\Gamma})$ are fully understood by studying ${\rm Cay}(\Gamma)$. This eventually motivated a question of A.~S. Kechris and A.~S. Marks: whether if $\Gamma$ and $\Delta$ are marked groups with isomorphic Cayley graph, then $ \chi_{B}( F(2^{\Gamma}))= \chi_{B}( F(2^{\Delta}))$; this is Problem 6.38 in the October 9, 2020 version of [``Descriptive graph combinatorics'', preprint, 2020]. This is known to be true when $\Gamma$ and $\Delta$ are finite. \par For $k\geq 3$, the author defines $\Gamma_{k} = {\Bbb{Z}/k\Bbb{Z}}\times\Bbb{Z}$ and $\Delta_{k} = {\Bbb{Z}/k\Bbb{Z}}\rtimes_{\varphi_{k}}\Bbb{Z}$ where ${\varphi\colon \Bbb{Z} \to {\rm Aut}(\Bbb{Z}/k\Bbb{Z})}$ is the homomorphism sending 1 to the inversion map. The main result of this paper establishes that $\chi_{B}( F(2^{\Gamma_{k}})) =k$ and $\chi_{B}( F(2^{\Delta_{k}})) = k+1$. Since $\Gamma_{k}$ and $\Delta_{k}$ have isomorphic Cayley graphs, this answers negatively the above question (the authors of [op. cit.] acknowledge this in an Addendum to Problem 6.38 in the October 9, 2020 version of the text). \par This is an interesting result and the paper is very well written. The author also provides a nice introduction to this field, and Sections 1 and 2 are particularly accessible to nonexperts. In Section 2 the author presents the elegant proof of the special case $k=3$. In Section 4 he discusses how the marked groups $\Gamma_{k}$ and $\Delta_{k}$ from above can be chosen with the additional requirement that the connected component equivalence relations on $F(2^{\Gamma_{k}})$ and $F(2^{\Delta_{k}})$ be isomorphic. Somewhat surprisingly, this argument uses the classification of hyperfinite Borel equivalence relations up to Borel isomorphism due to R.~L. Dougherty, S.~C. Jackson and Kechris [Trans. Amer. Math. Soc. {\bf 341} (1994), no.~1, 193--225 (Theorem 2$^\prime$); MR1149121]. In Section 5 the author discusses the same problems in the measurable setting. A list of further open questions is presented in Section 6.