Let $\Sigma$ be a finite alphabet, and consider the free monoid $\Sigma^*$. Given $w, w' \in \Sigma^*$ we say that $w$ overlaps $w'$ if there exist non-empty words $u, v, u'$ such that $w = uv$ and $w' = vu'$. Given a finite set of words $S$, define the overlap graph $OG(S)$ to be the simple graph with vertex set $S$ and an edge between two words in $S$ if one overlaps the other. In general this is defined as a weighted directed graph but in this case I do not care about direction or size of the overlaps.
I would like to understand better the cliques in $OG(S)$. In particular, I would like to know:
Given a large enough clique, can we find a large subclique with some nicer properties?
Is there a bound on the chromatic number $\chi(OG(S))$ in terms of the clique number $\omega(OG(S))$ and $| \Sigma |$?
These questions can be asked with some additional hypothesis on the words in $S$. In particular:
What if all words in $S$ are unbordered? A word $w$ is bordered if there exist words $u, v$ such that $w = uvu$ and $u \neq \epsilon$.
What if all words in $S$ are Lyndon words? A word $w$ is Lyndon for a given total order $\preceq$ on $\Sigma$ if it is primitive (i.e., not a positive power of another word) and smaller than all of its proper suffixes for the corresponding lexicographic order. All Lyndon words are unbordered.
Finally, what happens if we ask these question for the related graph $\overline{OG}(S)$, where we start from $OG(S)$ and add an edge between two words if one is a factor of the other? A word $w$ is a factor of a word $w'$ if there exist words $u, v$ such that $w' = uwv$.
All partial answers, as well as links to relevant literature, are welcome.