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 every edge red-green coloring of $\,K_s\,$ induces a red subgraph isomorphic to $\,K_5\,$ or green subgraph isomorphic to $\,K_{3\,3}\,$ ? Call this number $\,\mbox{RK}_{rg}.$

Q2 (double)  What is the least natural number $\,d\,$ such that every edge red-green coloring of $\,K_d\,$ induces a unicolored subgraph (totally red or totally gree) 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 edge red-green coloring of $\,K_d\,$ induces 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).