4. The professor has $n$ statements $\left(A_{1}, A_{2}, \ldots, A_{n}\right)$. He assigns dissertation topics to his graduate students: "prove that $A_{i}$ implies $A_{j}$." A dissertation should not be a direct logical consequence of previously defended works. What is the maximum number of graduate students the professor can have?
4. Answer: $n(n-1) / 2 + 1$.

Let's prove that more is impossible. First, note that if the number of dissertations in which statement $A_{i}$ was derived or, conversely, from which statement $A_{i}$ some other statement $A_{j}$ was derived is no more than $n$, then by discarding all such defenses, we get a situation with fewer statements, and the proof is completed by induction.

Thus, let for any $A_{i}$, the number of such dissertations is at least $n+1$. Consider a graph. Its vertices are $A_{i}$; an edge between $A_{i}$ and $A_{j}$ is drawn if both dissertations that connect them (i.e., in both directions) have been defended. Then each vertex is connected to at least two other vertices. (Otherwise, the number of dissertations involving statement $A_{i}$ would be no more than $n$.)

In this case, the edges form a cycle of some length $k$. This is impossible: an edge corresponds to a pair of opposite arrows, and among the $2k$ arrows (defenses of dissertations) corresponding to the edges of the cycle, there will be a last one in time. But then the topic of the corresponding dissertation would follow from previously defended ones. This leads to a contradiction.