>>10427674I don't think your argument will work because you're throwing away too much structure. What makes Ramsey-type theorems tick is the regularity of degrees; I cannot recall any theorem (in any part of graph theory) where dumping edges onto a graph arbitrarily accomplished anything.
The only way I know of doing this is slick, but requires you to know a couple of more elementary Ramsey results; you can either prove them, or find something uglier but more basic.
Pick a vertex v, and partition the 33 remaining vertices into subsets R,B,Y based on the colour of the edge to v. Since there is no blue K3 or red K3, R contains no red edges, so it is coloured blue/yellow with no blue triangle. If it has a yellow K4, done. Otherwise (here is the first ramsey result) R(3,4) = 9, so R has <= 8 vertices. Similarly, B <= 8.
This implies Y >= 33-16 = 17, but (here is the second ramsey result) we know R(3,3,3) = 17, so there is a monochromatic triangle in Y. But it's not red or blue so it's yellow, and it's joined to v by all yellow edges, and there's your K4.
