19.52 (2018)

The Gruenberg–Kegel graph (or the prime graph) $GK(G)$ of a finite group $G$ has vertex set consisting of all prime divisors of the order of $G$, and different vertices $p$ and $q$ are adjacent in $GK(G)$ if and only if the number $pq$ is an element order in $G$. Is there a finite non-solvable group $G$ such that $GK(G)$ does not contain 3-cocliques and is not isomorphic to the Gruenberg–Kegel graph of any finite solvable group?

It is known that there are no examples of such groups $G$ among almost simple groups.