Let $P_{i,j,k}=\{1_{i,j,k},\ldots,n_{i,j,k},\ldots,\gamma(i,j,k)_{i,j,k}\}$ (so that for each choice of $i$, $j$, and $k$, we have a disjoint set of size $\gamma(i,j,k)$). Let $Q=\{p_k\mid k\in K\}$ (so just $K$, up to relabeling). Let $R=\{\ast\}$. Let 
$$X=Q\coprod R\coprod_{\substack{i\in I,j\in J\\\k\in K}}P_{i,j,k}.$$
Define 
$$\Omega_j=\coprod_{i\in I,k\in K}P_{i,j,k}\subset X,$$
and 
$$f_{i}(n_{i_0,j_0,k_0})=\begin{cases}p_{k_0}\text{ if }i=i_0\\\ \ast\text{ otherwise}\end{cases}$$
$$f_i(p_k)=\ast\hskip0.3in f_i(\ast)=\ast$$
Thus $$f_i^{-1}(p_k)=\coprod_{j\in J}P_{i,j,k}\subset X,$$ and thus $f_i^{-1}(p_k)\cap \Omega_j=P_{i,j,k}$, so $|f_i^{-1}(p_k)\cap\Omega_j|=\gamma(i,j,k)$.