I am going through past papers because I am revising for my Graph Theory exam this week. I encountered the following question:
The bipartite Ramsey number $R(s,t)$ is the minimum $n$ s.t. a bipartite graph $G=(X \cup Y,E)$, when edge colored with ${\color{red}{red}}$ and ${\color{blue}{blue}}$, either has
1) A set $X^\prime \subseteq X$ of size $s$, $Y^\prime \subseteq Y$ of size $s$ for which all the edges between these sets are ${\color{blue}{blue}}$
2) A set $X^\prime \subseteq X$ of size $t$, $Y^\prime \subseteq Y$ of size $t$ for which all the edges between these sets are ${\color{red}{red}}$
Show that $$R(s,t) \leq (s+t)2^{s+t}. $$
Also: for each $n \geq 2$, there is a bipartite graph of size $n$ each such that for each $|X^\prime| = |Y^\prime| = 3 \log n$, there is a ${\color{red}{red}}$ and a ${\color{blue}{blue}}$ edge between $X^\prime$ and $Y^\prime$.
I have no clue how to even start with either statement... is there any one that can provide me with a hint?