The attatched is supposed to prove that $R(3,4)=9$ . One line say says there is no red $K_3$ in the two-colouring of $K_8$ What is it talking about?- I can see plenty of red triangles! (with corners which are the vertices of the graph of course). Unless I'm misunderstanding something...
Just switch Red and blue in the image, and this is a valid example. So just a mistake made by whoever made the image probably.