A simple graph is an ordered pair of sets $\,\Gamma:=(V\,E)\,$ such that $\,E\subseteq\binom V2.\ $ Kuratowski graph of the first kind is $\,K_n:=\left(V\,\,\binom V2\right),\,$ where $\,n:=|V|.\,$ And of the second type is
$$ K_{m\,n}\,:=\,(W\!\cup\!V\,\, \mu(W\times V)) $$
where $\,W\cap V=\emptyset,\,$ and $\,|W|=m\,$ and $\,|V|=n,\,$ and
$$\,\mu(W\times V)\,:=\,\{\{w\,v\}:\, (w\,v)\in W\times V\} $$
Q1 (single) What is the least natural number $\,s\,$ such that for every edge red-green coloring of $\,K_s,\,$ graph $\,K_s\,$ contains a red subgraph isomorphic to $\,K_5\,$ or a green subgraph isomorphic to $\,K_{3\,3}\,$ ? Call this number $\,\mbox{RK}_{rg}.$
Q2 (double) What is the least natural number $\,d\,$ such that for every edge red-green coloring of $\,K_d,\,$ graph $\,K_d\,$ contains a unicolored subgraph (totally red or totally greegreen) isomorphic to $\,K_5\,$ or to $\,K_{3\,3}\,$ ? Call this number $\,\mbox{RK}_{xx}.$
Q3 (triple) What is the least natural number $\,D\,$ such that every for every edge red-green coloring of $\,K_d,\,$ graph $\,K_d\,$ contains two unicolored subgraphs of the same color, one isomorphic to $\,K_5\,$ and the other one to $\,K_{3\,3}\,$ ? Call this number $\,\mbox{RK}_{YY}.$
REMARK Of course, the above questions allude to Kuratowski's characterization of planar graphs (the theorem covered even more than graphs).
We have $$ \mbox{RK}_{xx}\,\le\,\mbox{RK}_{rg}\,\le\,\mbox{RK}_{YY} $$
and, let me quote Taras Banakh,
$$ RK_{xx}\le R(5,5)\le 48 $$
There is a conjecture that $\,\,R(5,5)=43.\,$ (end of quote).
