Here’s the plan for Thursday for the Blue 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 \(k+1\).
- Henry: present exercise 1.10.
- Joe and Anthony: present these two exercises — show that if G is not connected, then its complement is connected, and show that a graph is bipartite if and only if it has no odd-length cycles.
- Nick: present exercise 1.12.
- Alex: present exercise 1.13.
- Ryan: present proof of Theorem 1.16.