Hint: consider any point on the graph. This point has $5$ protruding line segments, each of which are red or blue. We note that there must be at least $3$ line segments of the same color.
We can show that the points on the end of these $3$ segments form a red or a blue triangle. How?
Hint: Suppose that the three segments we found were red.
Let's look at the points on the end of those segments, and the connection from each to the other. If any two are connected with a red line, then we have a red triangle. Draw this to check.
Now, what if none of the two points at the end of these $3$ segments are connected with a red line? Can we say anything about either a red or blue triangle?
Some helpful pictures:
We looked at a point $A$, and noticed that whatever colors come out of $A$, we have to have $3$ that are the same
We then looked at the three points on the end. If there's a red line between any $2$ of them, then we have a red triangle. What happens if there isn't?
There must be a blue triangle