This is the excerpt from the book "Ramsey theory on the integers".
There are some confusing moments and I would be thankful if anybody helps me to grasp it.
The definition of $\nu_k(n)$ I understood. In the second paragraph author proves that $\nu_3(8)\leq 4$.
I have the following questions:
1) When considers the case $|T|\geq 5$ why in that case we are guaranteed to have a monochromatic 3-term AP?
2) Why he also working on the case $|T|\geq 6$? He already had worked on the case $|T|\geq 5$?!
3) Why he separately examining the case $|T|=5$?
First I will admit I do not fully understand this paragraph ... indeed I would have written ... There are $56$ subset of cardinality $5$ in $[1,8]$; I leave as an exercise to the reader to write them down & confirm that each one contains an arithemetic progression of length $3$. Hence blah blah.
This does not answer any of your 3 Questions.
1) This is a statement of what will be established & at this stage it is a claim.
2) He establishes that it suffices only to look at the case $ \mid T \mid =5$ because a larger cardinality would contain this case.
3) I have addressed in my answer to 2).