One wonders whether early 20th-century mathematicians Issai Schur [Jahresber. Deutsch. Math.-Ver. {\bf 25} (1916), 114--117; JFM 46.0193.02] and Bartel L. van~der~Waerden [Nieuw Arch. Wisk. {\bf 15} (1927), 212--216; JFM 53.0073.12] envisioned that their papers would spawn so much research into the following question: Given a partition of the natural numbers into finitely many color classes, what kinds of monochromatic patterns must appear? Schur showed that there must exist numbers $x$, $y$ and $z$ all with the same color such that $x+y=z$, and van~der~Waerden showed that there must be arbitrarily long monochromatic arithmetic progressions. The theory for linear patterns is well understood; see [N.~B. Hindman, in {\it Combinatorial number theory}, 265--298, de Gruyter, Berlin, 2007; MR2337052] for a survey. Much progress has been made for nonlinear patterns as well; see [M. Walters, J. London Math. Soc. (2) {\bf 61} (2000), no.~1, 1--12; MR1745405], for example, for polynomial generalizations and [A. Sisto, Electron. J. Combin. {\bf 18} (2011), no.~1, Paper 147; MR2817797] for an exponential one. This last result is that for every coloring of the natural numbers with two colors, there must exist $x>1$ and $y>1$ such that $x$, $y$ and $x^y$ have the same color. \par The first theorem of the important and well-written paper under review extends Sisto's result to any number of colors. To use the author's terminology, the set $\{x, y, x^y\}$ is partition regular. The second theorem tosses $xy$ into the monochromatic set as well. Other theorems go way beyond this to much more general patterns involving multiplication and exponentiation. In the other direction, the author shows that there exists a coloring for which one cannot find $x>1$ and $y>1$ such that $x$, $y$, $x^y$ and $y^x$ all have the same color, and goes on to characterize patterns that are partition regular in terms of graphs.