For any set $X$, let $[X]^2 = \{\{x,y\}:x\neq y\in X\}$. For $n\in\mathbb{N}$, let $[n]^2 = [\{1,\ldots,n\}]^2$. If $n\in\mathbb{N}$ is a positive integer and $S\subseteq [n]^2$, we set $$\chi(S) :=\min\{\kappa\in\omega\setminus\{0\}:(\exists f:\mathbb{N}\to\kappa)(\forall t\in S)\big|\text{im}(f|_t)\big| =2\}.$$ The expected chromatic number of a graph on $n$ vertices is defined by $$E_n:= \frac{1}{|{\cal P}([n]^2)|}\sum\{\chi(S):S\subseteq [n]^2\},$$ where of course $|{\cal P}([n]^2)| = 2^{n(n-1)/2}$.
What is the value of $\lim\inf_{n\to\infty}E_n/n$?
