Is there an easy way to deduce Ramsey's theorem of complete bipartite graph from the original Ramsey's theorem (including the hypergraph version)?
Ramsey's theorem of complete bipartite graph (the most basic version) states:
For every $m$ there exists $N$ such that every edge coloring $\chi:E[K_{N,N}]\to [r]$ contains a monochromatic subgraph $K_{m,m}$.
Let $r,m$ be given. By Ramsey's theorem there is a number $N$ such that every coloring $E(K_N)\to[r]$ contains a monochromatic $K_{2m}$.
I claim that any coloring $c:E(K_{N,N})\to[r]$ contains a monochromatic $K_{m,m}$.
Let the vertex sets of $K_{N,N}$ be $\{x_1,\dots,x_N\}$ and $\{y_1,\dots,y_N\}$. Consider $K_N$ with vertex set $[N]$. Define a coloring $\hat c:E(K_N)\to[r]$ so that, if $i,j\in N$ and $i\lt j$, then $\hat c(\{i,j\})=c(x_i,y_j)$. Let $i_1\lt i_2\lt\cdots\lt i_m\lt j_1\lt\cdots\lt j_m$ be such that $\{i_1,\dots,i_m,j_1,\dots,j_m\}$ induces a monochromatic $K_{2m}$ in $K_N$. Then $\{x_{i_1},\dots,x_{i_m}\}\cup\{y_{j_1},\dots,y_{j_m}\}$ induces a monochromatic $K_{m,m}$ in $K_{N,N}$.