For $S\subseteq\mathbb R^n$ a set
$C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of
$C$ is contained in $S$.
We show that there is essentially just one way to construct
a closed set $S\subseteq\mathbb R^2$ without an uncountable
$3$-clique that is not the union of countably many convex sets.
In particular, all such sets have the same convexity number;
that is, they
require the same number of convex subsets to cover them.
The main result follows from an analysis of the convex structure of closed
sets in $\mathbb R^2$ without uncountable 3-cliques in terms of
clopen, $P_4$-free graphs on Polish spaces.
convex cover, convexity number, continuous coloring, perfect graph, cograph
52A10 - Convex sets in $2$ dimensions (including convex curves) [See also 53A04]
03E17 - Cardinal characteristics of the continuum
03E75 - Applications of set theory