Given a $k$-fold multinomial distribution with equal probability for each category, the probability of the largest frequency in any category is desired. A simple asymptotic approximation to the upper percentage points of this distribution is obtained. A table of .95 and .99 points of the approximation for $k = 1(1)25$, and a table comparing these with actual values for $k = 3, 4, 5$ and $n = 3(1)12$, are provided. An investigation of the moment problem is given.
Robert M. Kozelka. "Approximate Upper Percentage Points for Extreme Values in Multinomial Sampling." Ann. Math. Statist. 27 (2) 507 - 512, June, 1956. https://doi.org/10.1214/aoms/1177728273