Call one vertex X. There are five others that we will divide into two sets.
Set R will be the vertices that are joined to X by a red edge.
Set B will be the vertices that are joined to X by a blue edge.
One of those set must contain at least three vertices. Say it is R.
If one pair of those three vertices is joined by a red side we are done. WHY?
If all three vertices are joined by blue edges we are also done. HOW?
How can you finish?