My question is, suppose we have $15$ vertices. What is the maximum number of edges we can add between these vertices, each with a fixed colored red or blue, so that there is no monochromatic triangle?
Since there's always a monochromatic triangle in a $K_6$, I think this problem amounts to finding the largest graph $G$ on $15$ vertices so that $\omega (G)=5$. However, I dont even know how to approach this. We can add $3$ edge-disjoint $K_5$'s on the 15 vertices...should I just try adding edges between them in a clever way to ensure the graph remains $K_6$-free?