For Tuesday, everyone should read up through the end of Section 3.1, and do all the exercises. In general when reading mathematics it is a good idea to have paper to hand, to draw pictures and verify claims. This section is one of those situations where it is not a good idea — it is absolutely essential.
At the end of last time, we saw the “AP focussing” argument used to show that \(W(3,2)\) exists. To make sure that we fully understand that argument before moving on, we should start on Tuesday by using the same argument to show that for each \(k\), \(W(k,2)\) exists, assuming that \(W(k-1,r)\) exists for every \(r\). I’ll present this (it is very quick — a valid upper bound is easy to justify using the same process that justified \(W(3,2)\leq 5\cdot 65\). The right picture fully explains the proof).
Next, we will move onto to the argument that \(W(3,3)\) is finite. This requires a new ingredient, because now we need to focus three monochromatic APs of length 2 (all of different colors) on a single, final focal point. Nick can present this.
The argument that \(W(3,3)\) is finite generalizes to show that \(W(3,r)\) is finite, for every \(r\). Sean can present this.
The last special case we’ll consider is \(W(4,3)\) (requiring focussing three monochromatic APs of length 3, all of different colors, on a single, final focal point). Alex can present this.
We should now be in a position to understand what’s going on in the proof of the general statement, that for each particular choice of \(k\) and \(r\), \(W(k,r)\) is finite (under the inductive hypothesis that \(W(k-1,r’)\) is finite for every \(r’\)). Henry can present this.