The graphs that I work with are all complete, each edge is colored red or blue, and each vertex is colored red or blue.
$\textbf{Definition:}$ A graph is $\textit{Happy}$ if there exists a vertex coloring such that each edge touches at least one vertex of its own color. $H(n)$ denotes the smallest size complete graph that is guaranteed to contain a happy $n$-graph (complete graph on $n$ vertices).
I want to find the exact value of $H(7)$, but am struggling to do so. I have previously found that $H(7) \leq 18$, know $6 \leq H(6) \leq 18$, and know that $H(6) \leq H(7)$. Please see http://mathurl.com/gmc8vwc for my formalized work for this. See https://i.stack.imgur.com/twW0J.jpg for a picture for $H(6) \leq 18$ argument.
This question was somewhat answered by @bof yesterday in this thread: https://math.stackexchange.com/q/1552279/80327 near the end of the comments section. How exactly does the person use known values of Ramsey numbers to say $H(7) \ge 18$?