4 added 531 characters in body; edited body

Here is a first second attempt :

Let (see edit history for previous version).

For each $P_{i,j,k}=\{1_{i,j,k},\ldots,n_{i,j,k},\ldots,\gamma(i,j,k)_{i,j,k}\}$ t\in\mathbb{N}$, let $$P_{i,j,k,t}=\{1_{i,j,k,t},\ldots,n_{i,j,k,t},\ldots,\gamma(i,j,k)_{i,j,k,t}\}$$ (so that for each choice of$i$, i\in I$, $j$, j\in J$,$k\in K$, and$k$, t\in\mathbb{N}$, we have a disjoint set of size $\gamma(i,j,k)$). Let

For each $Q=\{p_k\mid t\in\mathbb{N}$, let $$Q_t=\{a_{k,t}\mid k\in K\} K\}$$ (so for each $t\in\mathbb{N}$, this is just a copy of $K$, up to relabeling).

Let $R=\{\ast\}$. Let $$X=Q\coprod R\coprod_{\substack{i\in X=\coprod_{t\in\mathbb{N}}\left(Q_t\coprod_{\substack{i\in I,j\in J\\k\in K}}P_{i,j,k}.$$ K}}P_{i,j,k,t}\right).$$Define$$\Omega_j=\coprod_{i\in I,k\in K}P_{i,j,k}\subset K}P_{i,j,k,1}\subset X,$$and f_i:X\rightarrow X by$$f_{i}(n_{i_0,j_0,k_0})=\begin{cases}p_{k_0}\text{ $f_{i_0}(n_{i,j,k,t})=\begin{cases}a_{k,1}\text{ if }i=i_0\\ \ast\text{ i=i_0,t=1\\ n_{i,j,k,t+1}\text{ otherwise}\end{cases}f_i(p_k)=\ast\hskip0.3in f_i(\ast)=\ast$$f_i(a_{k,t})=a_{k,t+1}$$ Thus $$f_i^{-1}(p_k)=\coprod_{j\in J}P_{i,j,k}\subset X,$$ and thus$f_{i}^{-1}(n_{i,j,k,t})=\begin{cases}\emptyset\text{ if }t=1,2\\ \{n_{i,j,k,t-1}\}\text{ if }t>2\end{cases}f_i^{-1}(a_{k,t})=\begin{cases}\coprod_{j\in J}P_{i,j,k,1}\text{ if }t=1\\ \{a_{k,t-1}\}\text{ if }t>1\end{cases}$$We choose p_k=a_{k,1}. Thus f_i^{-1}(p_k)\cap \Omega_j=P_{i,j,k}, Omega_j=P_{i,j,k,1}, so |f_i^{-1}(p_k)\cap\Omega_j|=\gamma(i,j,k). Unfortunately this still doesn't address your size concerns, i.e. the preimage of any element of X being countable. Perhaps this can be fixed by adding more null elements , because if J is uncountable then f_i^{-1}(a_{k,1}) is uncountable (i.e. more I added the whole mess with the \ast's). t's to make the preimages of all the other elements countable). I'll leave this as a community wiki, and if anyone sees a way of fixing it they are welcome to edit this. 3 added 350 characters in body; [made Community Wiki] Here is a first attempt: Let P_{i,j,k}=\{1_{i,j,k},\ldots,n_{i,j,k},\ldots,\gamma(i,j,k)_{i,j,k}\} (so that for each choice of i, j, and k, we have a disjoint set of size \gamma(i,j,k)). Let Q=\{p_k\mid k\in K\} (so just K, up to relabeling). Let R=\{\ast\}. Let$$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,$$and f_i:X\rightarrow X by$$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,$$and thus f_i^{-1}(p_k)\cap \Omega_j=P_{i,j,k}, so |f_i^{-1}(p_k)\cap\Omega_j|=\gamma(i,j,k). Unfortunately this doesn't address your size concerns, i.e. the preimage of any element of X being countable. Perhaps this can be fixed by adding more null elements (i.e. more \ast's). I'll leave this as a community wiki, and if anyone sees a way of fixing it they are welcome to edit this. 2 added 19 characters in body Let P_{i,j,k}=\{1_{i,j,k},\ldots,n_{i,j,k},\ldots,\gamma(i,j,k)_{i,j,k}\} (so that for each choice of i, j, and k, we have a distinct disjoint set of size \gamma(i,j,k), one for each choice of i,j,k). \gamma(i,j,k)). Let Q=\{p_k\mid k\in K\} (so just K, up to relabeling). Let R=\{\ast\}. Let$$X=Q\coprod R\coprod_{i\in R\coprod_{\substack{i\in I,j\in J,k\in K}P_{i,j,k}.$$J\\k\in K}}P_{i,j,k}.$$ 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 $|f_i^{-1}(p_k)\cap\Omega_j|=\gamma(i,j,k)$.

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