All this discussion about the value of the Ramsey numbers is funny.
What is the ultimate value in the study of, e.g., numbers or algebraic varieties that is lacking in the study of elementary discrete structures, like graphs and hypergraphs?
Honestly, I do not have a motivated answer to this question, and a priori I cannot say that they do not deserve to be studied. There is actually some additional attraction, as for me, in the simplicity of these abstract objects.
What is their study indeed lacking, is the presence of deep, intricate and complicated structures, like the ones we encounter in the study of numbers, varieties or manifolds. This indeed can be said as making the study of the latter more appealing as the mathematics is exactly the study of hidden structures behind elementary abstract object; and most of the mathematics is naturally happening where there is the abundance of such structures. But I can not really say that such reasoning could be named by the vague term «value». At least, this does not seem motivated for me to be the reason for choosing numbers over graphs as more «valuable» objects of study.
(I am not a combinatorist myself, though I have some shallow passion to combinatorics lasting from the olympiad childhood.)

As many already said in this thread people in Hungarian combinatorics generally care about techniques and methods more than structures. The diagonal Ramsey problem is just one of the oldest and simplest problems that they haven't been able to settle after so many years of intensive effort and whose resolutions are believed to lead to major progress on other problems. Their mindset is different from that of algebraic geometers and number theorists as far as I know.

If you wanna talk about some "deep, intricate, and complicated structures" arising from graphs, then you're welcome to have a look at the so-called Graph Minor Theory of Robertson-Seymour. This is by no means Hungarian combinatorics but is enough to show you the study of graphs is definitely not lacking the presence of "deep, intricate, and complicated structures".