Here’s the plan for Friday for the Red group:
- Everyone: Read Section 1.3; do exercises 1.14 (easy), 1.18 and 1.20 (both moderate). You could also think about the following much harder variant of exercise 1.18: part of that exercise asks you to show that there is a sequence of reals of length \(k^2\) that has neither an increasing nor a decreasing subsequence of length \(k+1\). Show that this is best possible, that is, show that every sequence of reals of length \(k^2+1\) has either an increasing or a decreasing subsequence of length \(k+1\).
- Greyson: present exercise 1.8.
- Bailee: present a proof that the complement of a disconnected graph must be connected.
- Ted: present a proof that a graph is bipartite iff it has no odd cycle.
- Casey: present exercise 1.12.
- Colin: present exercise 1.13.
- Luca: present the proof of Theorem 1.16.