No valid $k$-partition of $[1,N]$ exists once $N \ge m$

This is the universal upper bound in its integer form.

Let $m \ge 2$ and $\ell, k \ge 1$, and suppose $N \ge m$. Then no family $P_0, \dots, P_{k-1}$ of subsets of $\mathbb{N}$ is a valid $k$-partition of $[1,N]$ into classes that are $\ell$-sum-free modulo $m$:

The obstruction is the single integer $m$ itself, which lies in $[1,N]$ and reduces to $0$ modulo $m$, so whichever class receives it is not $\ell$-sum-free.

Consequently $S_m(k,\ell) \le m-1$ for every $k$ and $\ell$, uniformly in both. This is the bound that makes the search space finite and justifies capping the definition of $S_m(k,\ell)$ at $m-1$.