Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent.
Suppose $\Gamma(V, E)$ is a finite simple graph. Let’s call a finite simple graph $\Gamma’(V’, E’)$ an induced subgraph of $\Gamma$ iff $V’ \subset V$ and $E’ = (V’ \times V’) \cap E$.
Let’s call a finite simple graph $\Gamma$ $n$-universal, iff any finite simple graph on $n$ vertices is isomorphic to some induced subgraph of $\Gamma$.
My question is:
What is the minimal $m$ for which $I_m$ is $n$-universal?
I already know, that $I_n^2$ is $n$-universal, as for any graph with $n$-vertices one can chose $n$ $n$-element subsets (indexed with vertices of that graph) of an $n^2$-element set, such that their their intersection is one-element if there is an edge between the corresponding vertices, and $0$ otherwise.
I also know, that $m$ should be larger that $n + o(1)$ as $I_m$ has $2^m$ vertices and the least possible $n$-universal graph has $(1 + o(1))2^n$ vertices