# Showing that $K_7$ contains at least 4 monochromatic triangles

A problem in my book is:

Let the edges of $K_7$ be colored with the colors red and blue. Show that there are at least four subgraphs $K_3$ with all three edges the same color (monochromatic triangles). Also show that equality can occur.

By the theorem on friends and strangers it is clear that 1 monochromatic triangle exists. Deleting a vertex of that triangle and applying the theorem again yields another. Why are two more guaranteed?

As an aside, a result in my book states that the number of monochromatic triangles in a 2-colored $K_n$ is at least $\binom{n}{3}-\lfloor \frac{n}{2}\lfloor (\frac{n-1}{2})^2 \rfloor \rfloor$. I want to demonstrate my solution without applying this result though as it appears later on in the book.

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A "biangle" is a triple of vertices $(a,b,c)$ where the edge joining $a$ to $b$ is not the same color as the edge joining $b$ to $c$. We call $b$ the "apex" of the biangle.