Apologies if the title is unclear, I couldn't think of a good way to express what I want.
Given the usual 2D representation of the complete graph $K_n$ as a regular $n$-gon with each pair of vertices joined, let $U_n$ be the set of the disjoint regions of space delimited by the edges of the graph (so $|U_3|=1, |U_4|=4, |U_5|=11...$). Further, label the nodes of $K_n$ as $p_1, ..., p_n$.
What is the size of the largest subset $S_n \subset U_n$ such that $\forall s \in S_n \exists i, j, k \in K_n$ with $s \subset \triangle ijk$ and $t \not\subset \triangle ijk \forall t \in S_n, t \ne s$? That is, How many points can I place inside drawing of $K_n$ such that each triangle formed by three points on the exterior of $K_n$ contains at most one point?
In case it helps clarify what I'm trying to do, I'm working on a puzzle where letters are encoded by giving three points from a discrete set defining a triangle in which the letter is written. For instance, for $K_4$:
1-------2 |\ A /| | \ / | | \ / | | X | | / \ | | / \ | |/ B \| 3-------4
"A" can be encoded as 123 or 124; B as 234 or 134. No further letters can be unambiguously encoded.
Playing around with pencil and paper, it seems as though $K_n$ can define $n-2$ letters unambiguously, but I haven't investigated far, and I can't immediately come up with a proof.