Gerhard has pointed out that your sharing-a-point graph is not universal for uncountable graphs, since any uncountable collection of functions on $\omega$ must have many of them sharing a point. So the graph with uncountably many points and no edges is not realized as an induced subgraph of your graph.

Meanwhile, since I find it interesting, let me point out that your graph is universal for countable graphs. That is, every simple graph on countable index set is realized as an induced subgraph. The reason is that we can build a copy of the countable random graph inside your graph, and the countable random graph is universal for countable graphs.

To build the countable random graph inside your graph, we successively choose functions $f_0$, $f_1$, $f_2$ and so on, in such a way that we realize every finite connection pattern. At stage $n$, we want to choose $f_n$ so that it is connected exactly to a certain finite set of previous $f_i$ and not to the others. By induction, we may assume that the previous functions are pairwise eventually pointwise unequal. So I may place finitely points into $f_n$ so as to realize the desired connectivity, and otherwise define $f_n(k)$ with totally new values. In this way, $f_n$ realizes exactly the desired connectivity to the previous functions. By iterating this and considering all possible finite connectivity patterns, the resulting graph will fulfill the finite pattern property. Since a back-and-forth argument shows that this property in a countable graph uniquely determines the countable random graph up to isomorphism, it follows that we have found a copy of the countable random graph in your graph.

Gerhard has pointed out that your sharing-a-point graph is not universal for uncountable graphs, since any uncountable collection of functions on $\omega$ must have many of them sharing a point. So the graph with uncountably many points and no edges is not realized as an induced subgraph of your graph.

Meanwhile, since I find it interesting, let me point out that your graph is universal for countable graphs. That is, every simple graph on countable index set is realized as an induced subgraph. The reason is that we can build a copy of the countable random graph inside your graph, and the countable random graph is universal for countable graphs.

To build the countable random graph inside your graph, we successively choose functions $f_0$, $f_1$, $f_2$ and so on, in such a way that we realize every finite connection pattern. At stage $n$, we want to choose $f_n$ so that it is connected exactly to a certain finite set of previous $f_i$ and not to the others. By induction, we may assume that the previous functions are pairwise eventually pointwise unequal. So I may place finitely points into $f_n$ so as to realize the desired connectivity, and otherwise define $f_n(k)$ with totally new values. In this way, $f_n$ realizes exactly the desired connectivity to the previous functions. By iterating this and considering all possible finite connectivity patterns, the resulting graph will fulfill the finite pattern property. Since a back-and-forth argument shows that this property in a countable graph uniquely determines the countable random graph up to isomorphism, it follows that we have found a copy of the countable random graph in your graph.

This copy of the countable random graph has the further nice property that two functions have at most one point in common and that common point is not used again by any other function. Thus, no three functions have a point in common.

Gerhard has pointed out that your sharing-a-point graph is not universal for uncountable graphs, since any uncountable collection of functions on $\omega$ must have many of them sharing a point. So the graph with uncountably many points and no edges is not realized as an induced subgraph of your graph.

Meanwhile, since I find it interesting, let me point out that your graph is universal for countable graphs. That is, every simple graph on countable index set is realized as an induced subgraph. The reason is that we can build a copy of the countable random graph inside your graph, and the countable random graph is universal for countable graphs.

To build the countable random graph inside your graph, we successively choose functions $f_0$, $f_1$, $f_2$ and so on, in such a way that we realize every finite connection pattern. At stage $n$, we want to choose $f_n$ so that it is connected exactly to a certain finite set of previous $f_i$ and not to the others. By induction, we may assume that the previous functions are pairwise eventually unequal. So I may place finitely points into $f_n$ so as to realize the desired connectivity, and otherwise define $f_n(k)$ with totally new values. In this way, $f_n$ realizes exactly the desired connectivity to the previous functions. By iterating this and considering all possible finite connectivity patterns, the resulting graph will fulfill the finite pattern property. Since a back-and-forth argument shows that this property in a countable graph uniquely determines the countable random graph up to isomorphism, it follows that we have found a copy of the countable random graph in your graph.

Gerhard has pointed out that your sharing-a-point graph is not universal for uncountable graphs, since any uncountable collection of functions on $\omega$ must have many of them sharing a point. So the graph with uncountably many points and no edges is not realized as an induced subgraph of your graph.

Meanwhile, since I find it interesting, let me point out that your graph is universal for countable graphs. That is, every simple graph on countable index set is realized as an induced subgraph. The reason is that we can build a copy of the countable random graph inside your graph, and the countable random graph is universal for countable graphs.

To build the countable random graph inside your graph, we successively choose functions $f_0$, $f_1$, $f_2$ and so on, in such a way that we realize every finite connection pattern. At stage $n$, we want to choose $f_n$ so that it is connected exactly to a certain finite set of previous $f_i$ and not to the others. By induction, we may assume that the previous functions are pairwise eventually pointwise unequal. So I may place finitely points into $f_n$ so as to realize the desired connectivity, and otherwise define $f_n(k)$ with totally new values. In this way, $f_n$ realizes exactly the desired connectivity to the previous functions. By iterating this and considering all possible finite connectivity patterns, the resulting graph will fulfill the finite pattern property. Since a back-and-forth argument shows that this property in a countable graph uniquely determines the countable random graph up to isomorphism, it follows that we have found a copy of the countable random graph in your graph.