\noindent $\textbf{Definition:}$ 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.\\ \\* \noindent Suppose we have an $18$-graph denoted by $G$. By Ramsey's Theorem, $G$ is guaranteed to contain a red $4$-clique or a blue $4$-clique. Let $G$ contain a red $4$-clique and denote it by $H$. Now $G$ is split into a $14$-graph denoted as $G'$ and $H$. Again by known values of ramsey numbers, $G'$ is guaranteed to contain a red $3$-graph or a blue $5$-graph. Let $G'$ contain a blue $3$-clique and denote it $H'$. Then, $G$ contains $H$, $H'$, and $G'$. Since $H$ and $H'$ are subgraphs of the complete graph $G$, both $H$ and $H'$ are fully connected to each other. The graph formed by these two subgraphs have size $|H|+|H'| = 7$ and is a happy graph since it is partioned into two cliques of distinct color. Therefore, an $18$-graph is guaranteed to contain a $7$-happy graph. Observe that we have shown that a $7$-happy graph is guaranteed to be contained in a $18$-graph. Hence, \[ H(7) \leq 18 \] where $18$ is the new smallest upperbound for the happy number of a $7$-happy graph. \\ \noindent Now consider $G$ so that it is comprised of $G'$ and $H$. Let $G'$ contain a blue $5$-graph instead of a red $4$-graph as previously done above and denote the blue $5$-graph as $H'$. Let $v$ be a vertex of $H$ and note that it is connected to all of $H'$ since both graphs are induced from a complete graph. The graph made by $v$ and $H$, denoted as $V$, forms a $6$-graph which is partioned into a red clique of size $1$ and a blue clique of size $5$. Therefore, $V$ is a $6$-happy graph and is contained within a $18$-graph. Hence, \[ H(6) \leq 18 \] where $18$ is the new smallest upperbound for the happy number of a $6$-happy graph. - - - - - \noindent Question remains: What is the exact value of $H(7)$?