Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules.

1. [2.1.{25,26}] Recall that the slope of the line segment joining the pair of points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane is $(y_2-y_1)∕(x_2-x_1)$.

   1. Prove that if $S$ as a set of 17 points in the plane, no two of which are on a common vertical or horizontal line, then there exist $p_1,\dots ,p_5\in S$ such that the slope of the line segments joining $p_i$ and $p_j$ for $1\le i<j\le 5$ all have the same sign.
   2. Give an example that shows that the conclusion of part (a) does not always hold if we assume only that $|S|\ge 16$.
2. [2.2.3] What is the maximum number of edges in an $n$-vertex bipartite graph? Prove your answer is correct.
3. [2.2.6] Each of 9 users sends three friend requests on a social media platform. Is it possible that each person $p$ receives exactly 3 friend requests from the same three people to whom $p$ sent the requests? What if the number of users is 8 instead?

4. [2.2.13] Show that the two graphs below are isomorphic.