With $r(G)$ I refer to the smallest $n$ such that every blue-red colouring of the edges of $K_n$ contains a monochromatic copy of the grpah $G$ (this exists because $r(G)\le R(|G|)$).
Now let $I_k$ be a set of $k$ independent edges, so $|I_k|=2k$.
My question now is, why is $r(I_k)=3k-1$ ?
Consider the complete graph on the vertex set $V=\{1,2,\dots,3k-2\}$. Color the edge $xy$ red if $\max\{x,y\}\le2k-1$, blue otherwise. Clearly, there is no set of $k$ independent edges of the same color. This shows that $r(I_k)\ge3k-1$.
We prove $r(I_k)\le3k-1$ by induction on $k$. The case $k=1$ is clear, so assume $k\ge2$. Consider the complete graph on a vertex set $V$ with $|V|=3k-1$, suppose each edge is colored red or blue. If all edges have the same color, we are done. If the edges are not all the same color, then there are two adjacent edges of different colors; say $xy$ is red and $xz$ is blue. By the induction hypothesis, the subgraph induced by $V\setminus\{x,y,z\}$ contains $k-1$ independent edges of one color; depending on the color of those $k-1$ edges, adjoining $xy$ or $xz$ will produce a set of $k$ independent edges of one color.