Some comments from Jose

1. When you are proving that k is the minimum number with respect to some property (like in the chromatic number problems), the proof has to consist of two parts. The first one is showing that k satisfies the property. The second one is showing that it actually is the minimum, that is, you have to show that the numbers smaller than k do not satisfy the property!!! Both steps are crucial!
2. Many of you stated that the chromatic number of a graph G is the maximum n such that K_n is a subgraph of G. This is in general NOT true. There are graphs with large chromatic number that do not contain even a copy of K_3: for any positive m there is a graph G_m whose chromatic number is at least m and such that G_m contains no K_3 as a subgraph. Producing explicit examples of this type is usually not easy, but there are methods coming from probability (some of which we will discuss later this quarter) that allow us to show the existence of such beasts. To summarize, the chromatic number of a graph G is *at least as large as* (but not necessarily equal to) the maximum n such that K_n is a subgraph of G.
3. Remember that there are some counting problems where you have to keep track of the order in which you make choices and then eliminate redundancies. This often happens when you are counting permutations with a given cycle structure. DON'T FORGET THAT.