I am covering Ramsey Theory at this time while learning Combinatorics and I needed clarification on something. Any help would be appreciated.
In the text, it states the following: "We show that r(2,n) = n:
- r(2, n) $\le$ n: If we color the edges of $K_n$ either red or blue, then either some edge is colored red (and so we have a red $K_2$) or all edges are blue (and so we have a blue $K_n$).
- r(2, n) > n - 1: If we color all the edges of $K_{n-1}$ blue, then we have neither a red $K_2$ nor a blue $K_n$." Since r(2,n)$\le$ n and r(2,n) > n-1, r(2,n) = n.
$K_n$ is a complete graph on n vertices. What I am confused about is the second part, if you color all the edges of $K_{n-1}$ blue, then you might not have a red $K_2$, but you have a blue $K_{n-1}$, and say for n-1 $\ge$ 2, the blue $K_{n-1}$ would contain a blue $K_2$ as a subgraph. And since r(2,n)=r(n,2), it shouldn't matter what we call blue and what we call red, it's arbitrary. Maybe there's a flaw in my understanding but I would appreciate it if anyone could clarify this.