15.41 (2002)

Let $R(m, p)$ denote the largest finite $m$-generator group of prime exponent $p$.
$\qquad$ a) Can the nilpotency class of $R(m, p)$ be bounded by a polynomial in $m$? (This is true for $p = 2, 3, 5, 7$.)
$\qquad$ b) Can the nilpotency class of $R(m, p)$ be bounded by a linear function in $m$? (This is true for $p = 2, 3, 5$.)
$\qquad$ c) In particular, can the nilpotency class of $R(m, 7)$ be bounded by a linear function in $m$?

My guess is “no” to the first two questions for general $p$, but “yes” to the third. By contrast, a beautiful and simple argument of Mike Newman shows that if $m \geqslant 2$ and $k \geqslant 2$ ($k \geqslant 3$ for $p = 2$), then the order of $R(m, p^k)$ is at least $p^{p^{\cdot^{\cdot^{\cdot^{p^m}}}}}$, with $p$ appearing $k$ times in the tower; see (M. Vaughan-Lee, E. I. Zelmanov, J. Austral. Math. Soc. (A), 67, no. 2 (1999), 261–271).