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 $|T|=5$?