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We study the asymptotic behavior of the maximal multiplicity$M_n=M_n(\sigma)$ of the blocks in a set partition of $[n]=\{1,2,...,n\}$,assuming that $\sigma$ is chosen uniformly at random from the set of all suchpartitions. Let $W=W(n)$ be the unique positive root of the equation $We^W=n$and let $f_n$ be the fractional part of $W(n)$. Furthermore, let$R_n=W^{\lfloor W\rfloor}/\lfloor W\rfloor !$ and let$\vartheta_n=\min{\{f_n,1-f_n\}}$. We show that, over a subsequence$\{n_k\}_{k\ge 1}$, $(M_{n_k}-R_{n_k})/\sqrt{R_{n_k}}$ converges weakly, as$k\to\infty$, to $\max{\{Z_1,Z_2-u\}}$, where $Z_1$ and $Z_2$ are twoindependent copies of a standard normal random variable and either$u=\left(\frac{1}{2\pi}\right)^{1/4}\lim_{k\to\infty}\vartheta_{n_k}\frac{\sqrt{n_k}}{\log^{7/4}{n_k}}\in[0,\infty)$ or $u=\infty$. The proof uses the saddle point method. A comparisonwith the similar statistic for random integer partitions of $n$ is also given.

On the maximal multiplicity of block sizes in a random set partition