Optimal Thresholds for Monotone Non-Boolean Functions

Let $[q] = \{0,1,\ldots,q-1\}$, let $\Delta[q]$ denote the simplex of probability measures on $[q]$, and let $\gamma$ denote the Lebesgue measure normalized on $\Delta[q]$. We prove that for any symmetric monotone function $f \colon[q]^n \to [q]$ and any $a \in [q]$ we have \begin{equation*} \gamma(\{\mu \in \Delta[q]\;\vert\;\mathbb{P}_{x\sim\mu^{\otimes n}}[f(x)=a] \in (\varepsilon,1-\varepsilon)\}) = O(1/\log n)\text{.} \end{equation*} We also show that this bound is tight. This improves Kalai and Mossel's previous bound of $O(\log \log n/\log n)$ and answers their question completely.