Take $N$ examples sampled from a multinomial distribution $(p_1,p_2,\cdots,p_m)$ with $m$ outcomes, with $N_i$ ($1 \leq i\leq m$) being the number of examples taking the outcome $i$. Here I assume $p_1,p_2,\cdots,p_m$ are listed in the descending order; if not, we can reorder them to make them descending. My objective is to make the most frequently occuring outcome $I_{max}=\arg \max \limits_i N_i$ as close to 1 as possible.
So my question is what size of $N$ we should take to make $I_{max}\leq \tilde{m}$ for some $1\leq \tilde{m}\leq m$? Furthermore, how $N$ determines $I_{max}$ ? The tool I can think might be useful is Bretagnolle-Huber-Carol inequality.
Any clues or references are appreciated. Thanks a lot in advance.