In this paper, for \(k\in \mathbb {N}\) , we introduce special sets \(\mathcal {A}^{ksmc}\) , referred to as k semi-colored sets. In particular, for \(k\ge 2\) , we construct such sets for which, beyond a certain threshold, the partition function \(p_{{\mathcal {A}}^{(k-1)smc}}(n)\) , i.e. the number of partitions of n with parts in \({{\mathcal {A}}^{(k-1)smc}}\) , is congruent to 0 modulo k. Furthermore, we investigate “thin” sets characterized by the counting function \(\mathcal {A}^{(k-1)smc}(x)\asymp \log x\) as \( x\longrightarrow \infty \) . In contrast, we also provide two examples of “dense” sets, where the counting function satisfies \(\mathcal {A}^{(k-1)smc}(x)\gg \frac{x}{\log x}\) as \(x\longrightarrow \infty \) .
