If R(5,5)=48, each vertex must be (23,24) colored.
Proof:
Suppose vertex X is not (23,24) colored.
Then there must be a color A used >=25 times. Denote the other color B.
Denote the set of vertices connected with X by color A as Y. By definition, |Y|>=25.
Thus Y must contain a K4 with color A, or a K5 with color B. Either of them contradicts R(5,5).
Proof:
Suppose vertex X is not (23,24) colored.
Then there must be a color A used >=25 times. Denote the other color B.
Denote the set of vertices connected with X by color A as Y. By definition, |Y|>=25.
Thus Y must contain a K4 with color A, or a K5 with color B. Either of them contradicts R(5,5).
