Reference Request: accessible points of Wada domain boundaries in $\mathbb R^d$

Question

Say that a compact $K\subset\mathbb R^d$ has the Wada property if there are disjoint, connected open sets $V_1,V_2,\ldots,V_n\subset\mathbb R^d$, $n\geq3$, such that $K=\partial V_1=\partial V_2=\ldots=\partial V_n$. In conversation with someone, they mentioned something along the lines of: in the plane, no point of such a $K$ (or perhaps it was 'not many points' in the sense of Hausdorff dimension) could be accessible from all of the $V_i$ simultaneously, but for $d>2$, this need not be the case. I haven't been able to find any sources on Wada domains coming close to this claim (most papers I have seen seem to be preoccupied with $d=2$, and with physical applications rather than the geometry of $K$). Does anyone have a good source for this, or more generally for the theory of accessible points in Wada domain boundaries?

Answer 1

For $n=2$, suppose that three domains $D_j$ have three points $A_j$ accessible from each. Let us choose three points $B_j\in D_j$ and for each $j$ connect each $B_j$ to each $A_k, k=1,2,3$ by three disjoint curves within $D_j$. These 6 points and 9 curves make a graph which is known as $K_{3,3}$ in graph theory, and it is a famous kindergarten puzzle to show that this graph is not planar.

This shows that three plain domains cannot have 3 points accessible from all three of them.

If $n\geq 3$ there is much more space, of course, and it is easy to see that any finite graph can be embedded. So you can have any number of regions with any number of points accessible from all of them.