Here are some bounds that I can extract from the dynamic survey [Small Ramsey Numbers][1] by Stanisław Radziszowski. Recall that for two graphs $G$ and $H$, $R(G,H)$ is the smallest integer $n$ such that every red-green edge colouring of $K_n$ contains a red $G$ subgraph or a green $H$ subgraph.  In this notation, $\mathrm{RK}_{rg}=R(K_5, K_{3,3})$.  

**Claim.** 
$13 \le \mathrm{RK}_{xx} \le 18.$

*Proof.* The lower bound $13 \le \mathrm{RK}_{xx}$ was proven by Will Brian in the  above comments.  For the upperbound, we have 
$\mathrm{RK}_{xx} \le R(K_{3,3}, K_{3,3}) = 18$.  Note that $R(K_{3,3}, K_{3,3})=18$ was proven by H. Harborth and I. Mengersen in *The Ramsey Number of $K_{3,3}$* (see Section 3.3.1 of the survey). $\Box$

**Claim.** $21 \le \mathrm{RK}_{rg} \le 47$.

*Proof.* For the lowerbound, observe that $K_{5,5,5,5}$ does not contain $K_5$ and the complement of $K_{5,5,5,5}$ does not contain $K_{3,3}$.  For the upperbound, consider the edges incident to a fixed vertex. This gives the easy inductive bound $\mathrm{RK}_{rg} \le R(K_5, K_{2,3})+R(K_4, K_{3,3})+1$.  Repeating the argument, we obtain 
$$\mathrm{RK}_{rg} \le R(K_5, K_{2,3}) + R(K_4, K_{2,3})+R(K_3, K_{3,3})+2.$$ In Section 5.9 of the survey, we have $R(B_3, K_4)=14$ and $R(B_3, K_5)=20$, where $B_3=K_2 + \overline{K_3}$.  Since $K_{2,3} \subseteq B_3$, we have $R(K_5, K_{2,3}) \le 20$, and $R(K_4, K_{2,3}) \le 14$.  Finally, in Section 3.2 of the survey, it is noted that $R(K_3, G)$ has been computed exactly for all connected graphs up to $9$ vertices.  The value of $R(K_3, K_{3,3})$ is not given explicitly in the survey, but tracking down the references, we have $R(K_3, K_{3,3})=11$.  Substituting, we obtain $\mathrm{RK}_{rg} \le 47$, as required. $\Box$ 

Finally, here is an upperbound for $\mathrm{RK}_{YY}$.

**Claim**. $\mathrm{RK}_{YY} \le 70$.  

*Proof.* Since $K_6-e$ contains both $K_5$ and $K_{3,3}$, we have $$\mathrm{RK}_{YY} \le R(K_6-e, K_6-e) \le 70$$ (see Section 3.1 of the survey). $\Box$



  [1]: https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1