Given a simple graph $G$ [no loops or parallel edges] on six vertices, let $G^c$ denote its complement. It is known that either $G$ or $G^c$ must contain a triangle $T$ in it. [An example of a Ramsey number I think.]
My question came up when I tried to find such a graph on six vertices which had only one $T$ in it or its complement. I believe if I did things right there are none of these. This I tried to check by drawing a copy of each graph $G$ on six vertices which has a $T$ in it, and then looking at the complements to see if any had no $T$'s in them. I found none.
One way to pose my question for a general number $n$ of vertices is then as follows: Consider the set of all graphs $G$ on $n$ vertices, and for each one construct $G^c.$ Count the number of $T$ in $G$ and add that to the number of $T$ in $G^c.$ Finally find the minimal such sum for the given $n$ and call that say $M(n).$ So my above six vertex conjecture is that $M(6)=2.$
By considering a cycle graph on five vertices we find $M(5)=0.$
So going further one could ask if anything is known about this minimum function $M(n)$ for larger $n$ [such as bounds, etc.] And does it have a name?
I would also like to see a simpler way to handle the six vertex case, one not involving drawing all the graphs.