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:

$R(3,8)=28$ | Brendan McKay & Zhang Ke Min, The Value of the Ramsey Number R(3,8).

$R(3,9)=36$ | Charles Grinstead & Sam Roberts, On the Ramsey Numbers R(3,8) and R(3,9)

$R(4,5)=25$ | Brendan McKay & Stanislaw Radziszowski, R(4,5)=25.