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In this paper, we study the relationship between the number of $n$-vertex graphs in a hereditary class $\cal X$, also known as the speed of the class $\cal X$, and boundedness of the clique-width in this class. We show that if the speed of $\cal X$ is faster than $n!c^n$ for any $c$, then the clique-width of graphs in $\cal X$ is unbounded, while if the speed does not exceed the Bell number $B_n$, then the clique-width is bounded by a constant. The situation in the range between these twodoi:10.37236/124 fatcat:t56dabntpjdibcp5znwj2e3u7q