= Solution
With <existential rules> understood as function-free <tuple-generating dependencies>, $\boxed{\text{only (ii)}}$ belongs to the fragment. Split its antecedent to obtain
$$
\begin{aligned}
\mathit{Undergrad}(x)&\to\exists y\,\mathit{SupervisedBy}(x,y),\\
\mathit{Masters}(x)&\to\exists y\,\mathit{SupervisedBy}(x,y).
\end{aligned}
$$
These two <existential rules> have exactly the same meaning as (ii): either category requires a supervisor, and when both hold the same supervisor may witness both requirements.
Sentence (i) requires a genuine choice between two positive alternatives and also an exclusion condition. Positive <existential rules> cannot enforce that exclusion: interpreting all their relations as full relations on a one-element domain always satisfies them, whereas that interpretation violates (i). Sentence (iii), in the supplied vocabulary, uses the nonconstant function $\mathit{CountSupervise}$ and is therefore outside the function-free fragment. Introducing a new relation for the counting comparison would change the representation and require additional axioms; it is not an equivalent sentence in the stated vocabulary.
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