One colour: $S_m(1,\ell) = \min(\ell-1, \lfloor m/\ell \rfloor)$ for $2 \le \ell \le m$

This is the exact single-colour modular Schur number in the range the literature posed it for.

For every $m \ge 2$ and every $\ell$ with $2 \le \ell \le m$,

The two terms are the two ways a single class can fail. The term $\ell - 1$ is the no-wrap regime, where the class is defeated by $\ell$ copies of $1$ summing to $\ell$; the term $\lfloor m/\ell \rfloor$ is the wrap regime, where some $\ell$-tuple sums past the modulus and lands back at $\overline{1}$. Which one binds depends on whether $\ell(\ell-1) < m$.

This is one of the two regimes of the problem that admit an exact formula, the other being the many-colours regime of the main closed form; the intermediate range $1 < k < n-1$ has no known formula.