**Definitions**: For a graph $G$ we denote the independence number by $\alpha(G)$ and the clique number by $\omega(G)$. A graph is vertex-transitive if for every pair of vertices $x,y \in V(G)$ there is an automorphism that takes $x$ to $y$.

**Question**: What is the order of magnitude of the minimum of $\alpha(G)\times \omega(G)$ for vertex-transitive graphs?

**Motivation/Additional information**: I was thinking about a conjecture of János Körner. Let us say that two permutations have a flip if there are two coordinates that contain the same two elements in both permutations but the order of the two elements is different. Körner conjectures that the maximal size of a set of permutations where each pair have a flip is only exponential in the length of the permutations. One way to look at this problem is as follows. Define a graph $H$ where the vertices are the permutations, and we connect two vertices if the corresponding permutations have a flip. Then we have $|V(H)|=m!$. Körner conjectures that $\omega(H)$ is only exponential in $m$. Josef Cibulka determined the order of magnitude of the independence number of this graph: $\alpha(H) \sim m^ \frac{m}{2} $ (The construction is very nice, he calls a "flip" a "reverse", the paper can be found here https://pdfs.semanticscholar.org/aa5f/7de6b1727bb7618d6674f2aa848bf69335c6.pdf). The following inequality holds for all vertex transitive graphs, hence also for $H$: $$\alpha(H) \times \omega(H) \leq |V(H)|.$$

If the conjecture of Körner is true then $\alpha(H) \times \omega(H) \sim m^{\frac{m}{2}-o(m)}$ and $|V(H)|\sim m^m$. If we normalize the number of vertices of $H$ to be $n$ (as usual), we have a vertex transitive graph where $\alpha \times \omega$ is as small as $n^{\frac{1}{2}-o(1)}$. But I could not construct such a graph.

For not necessarily vertex transitive graphs $\alpha(G)\times \omega(G)$ can be as small as $\log(n)^2$, an example is an Erdős-Rényi random graph for $p=1/2$. (I am not sure whether this is indeed the minimum for general graphs, but this is not relevant for the conjecture that I am interested in.)

I could construct vertex-transitive graphs where $\alpha \times \omega \sim n^{c-o(1)}$ for $c \in (1/2,1)$ as follows: The pentagon has $\alpha \times \omega=4$ and $n=5$. We can blow up a graph as follows. If $G$ has $n$ vertices, take $n$ vertex disjoint copies of $G$. Fix an arbitrary pairing of these vertex disjoint copies with the vertices of the original graph. And if two vertex disjoint copies have their pairs connected in $G$, then add every edge between these disjoint copies. It is easy to see that if $G'$ is the blowup of $G$, then $\alpha(G')=\alpha(G)^2$ and $\omega(G')=\omega(G)^2$ and $|V(G')|=|V(G)|^2$ and if $G$ was vertex-transitive, $G'$ is also. Thus blowing up the pentagon many times we get a sequence of graphs where (after normalizing the number of vertices) we have $\alpha \times \omega \sim n^{0.8613}$.