Consider the complete graph k₁₆ and group the vertices into 4 groups of size 4 . Colour an edge between two vertices red if they are in the same group, and colour it blue otherwise. Show that this edge colouring does not contain any monochromatic 5cliques, i.e., r(5,5)≥17.
Remark: It seemed the first strategy is bad compared to the second. However, when k is large, the first strategy gives r(k,k)>2k/2while the second gives r(k,k)>(k-1)².
