The reason for the gorgeous simplicity of the classic proofs of $R(3,3)$, $R(4,4)$, $R(3,4)$ and $R(3,5)$ is that essentially all you need is the trivial bound and a picture.
But for bigger Ramsey numbers, the bound $R(p,q)\leq R(p-1,q)+R(p,q-1)$ is not best anymore, which adds to the complication of having an increasingly intractable graphs involved.
For $R(3,6)$ and $R(3,7)$, you can replace it with another simple bound, and the construction of an explicit coloring does the rest.
(Actually, I don't think I've seen a proof of $R(3,7)\leq 23$. The original paper by Kalbfleisch proves that $R(3,7)\geq 23$, and mentions a communication by Jack Graver and Jim Yackel at a conference for the other inequality. Yet Kalbfleisch is usually credited with the proof (see here por example))
(Actualization. As j.c. mentions in the comments, the $R(3,7)\leq 23$ bound by Graver and Yackel was published two years, see the paper here, page 21, computation O (the notation is shifted by 1)).
In comparison with those, the proof of the other known Ramsey numbers, $R(3,8)$, $R(3,9)$ and $R(4,5)$ seem extremely unenlightening. Obviously the size of the colorings make neccesary extensive computation, so perhaps the proofs are as good as possible. But still I feel like I should ask.
Is there a simpler or more intuitive proof of any of those three results?
Since all three original papers are avaible online, I'll include them for easier reference: