  This paper studies the choice number and paint number of the lexicographic
product of graphs. We prove that if $G$ has maximum degree $\Delta$, then for
any graph $H$ on $n$ vertices $ch(G[H]) \le (4\Delta+2)(ch(H) +\log_2 n)$ and
$\chi_P(G[H]) \le (4\Delta+2) (\chi_P(H)+ \log_2 n)$.
