Solution:

We claim the only way for Purineqa to deduce Arno's preferences is for each pizza to contain exactly one topping, with no topping be repeated. It is obvious that she can deduce the toppings in this case.

We now claim that this is not possible with any other combination. Suppose that Arno tells Purineqa that he does not like any of the five pizzas. Then, Purineqa should be able to rule out at least one of the possibilities that Arno likes none of the toppings and that Arno likes exactly one of the toppings $T$. It is clear that this is possible if and only if there is a pizza with only $T$ on it. This is true for all five toppings $T$, so we're done.