I'm trying to understand a proof:
$R(3,3) = 6$
Take a red/blue colouring of $K_6$. Take a vertex $v$ (is an element of) $V(K_6)$, either $v$ is incident to $\geq 3$ red edges or, $v$ is incident to $\geq 3$ blue edges.
Without loss of generality, the former holds. Consider the other endpoints of these edges, either $2$ of them are joined by a red edge - so we have a red $K_3$ - or they are all joined by blue edges and we get a blue $K_3$. Hence $R(3,3)=6$.
Now, this is simple enough, but what is confusing me is the construction of the edge colouring. I thought that, like vertex colouring, we coloured the edges so that no two adjacent edges had the same colour. But here, in the first part of the proof to obtain a red $K_3$, adjacent edges are coloured red, and in the second part of the proof adjacent edges are coloured blue!
Could someone please explain this to me, maybe I am missing a tidbit of vital knowledge of the rules of edge colouring?