Hint:
Given a partial coloring of the graph K_n, for any subgraph $K$ of size $4$
(so $K$ is a copy of $K_4$),
define a weight function w(K) as follows: if at least one
edge of K is red and at least one edge is blue, then w(K)=0.
If no edge is colored, then w(K)= 2^{-5}. If r>=1 edges
are colored all with the same color, then w(K)=2^{r-6}.
Define W = sum over all K which are copies of K_4 in K_n
of w(K).
Note that w(K) is the probability that $K$ will be monochromatic
if the remaining edges are assigned colors randomly and independently.
Thus W is the expected number of monochromatic copies in a
random extension of the partial coloring to a full coloring.