35. In a group of 30 people, each person likes exactly $k$ people in the group. What is the minimum $k$ for which we can assert that there will definitely be two people who like each other?
75.35. For $k=15$. Indeed, if we consider a graph with 30 vertices, each of which is connected by an edge, and place an arrow on edge $A B$ from $A$ to $B$ if person $B$ likes person $A$, then we should place $30 \cdot 15=$ $=450$ arrows, while the total number of edges is $30 \times$ $\times 29 / 2=435$. Therefore, on some edge, two arrows will appear (of course, in opposite directions). An example for $k<15$ is constructed as follows: arrange all people in a circle, after which we will assume that each person likes exactly $k$ people following them in a clockwise direction.