I hope this is close to what you are asking. The following compactness principle turns out to be useful in certain construction in dynamical systems and in probability (in particular, in the theory of exchangeable random variables), and it may be seen as a topological version of the infinitary Ramsey theorem.
Lemma. Let $X,d$ a compact metric space, and
$u:\mathbb{N}^2\to X$ a double
sequence in $X$. Then, there exists a
strictly increasing
$\sigma:\mathbb{N}\to\mathbb{N}$ such
that, denoting
$v(i,j):=u(\sigma(i),\sigma(\, j)),$
the following limits exist, and
coincide:
$$\lim_{i\to\infty} \lim_{j\to\infty} v(i,j) \, = \lim_{ i< j, \,(i,j)\to\infty} v(i,j). $$
The proof is just routine (iterated) application of the usual diagonal argument for sequences. How does it implies the infinitary Ramsey theorem? Take $X$ a discrete space of colors and $u$ an
$X$-coloring of the complete graph with vertices set $\mathbb{N}$. Then the existence of the limit in the RHS means that the set $\{\sigma(i):i> c\}$ for some c is a monocromatic complete subgraph.
One can even state a more general version for multi-sequences $u(i)$ indicized on increasing $n$-ples $i:=(i_1< i_2\dots < i_n)$ of natural numbers; the game is that any parenthesization
$$(i_1,\dots,i_{k_1}),(i_{k_1+1},\dots,i_{k_2}),\dots,(i_{k_r},\dots,i_n)$$
produces a different iterated way of letting $i$ go to infinity (like when making the beads sliding from left to right in an abacus: in small clusters, one at a time, or all together ).
The corresponding limits for u(i) may or may not exist and/or coincide; but, up to a selection of indices via a strictly increasing $\sigma:\mathbb{N}\to\mathbb{N}$, all these iterated limits do exist and coincide.
Actually, it may be argued whether it really gives a fundamentally different proof of Ramsey theorem as you are asking. Nevertheless, if you try submitting it to an analyst or to a geometer, I think you are much more likely to obtain an immediate proof of it than with the original set theoretic version (please confirm my guess). On the other hand, it may sound quite weird to a pure set theorist (do not necessarily confirm this statement).
The best bounds I know of are due to Tom Bohman for $R(k,4)$ and Bohman and Peter Keevash for $R(k,5)$ and beyond. Both rely on using the differential equations method to analyze the following process: Start with the empty graph, and at each step add an edge uniformly at random among all edges which do not create a $K_t$. The bounds they achieve are
$$R(k,t) \geq c_t \left( \frac{k}{\log k} \right)^{\frac{t+1}{2}} (\log k)^{\frac{1}{t-2}}$$
The final term in this product corresponds to the improvement over the bounds obtained using the Local Lemma. For $t=3$ it matches Kim's bound up to a constant factor.
Best Answer
The best known explicit Ramsey graph construction is in the paper:
Call a graph $K$-Ramsey if it doesn't have a $K$-clique or a $K$-independent set. They prove
Here, "explicit construction" means roughly that there is an efficient algorithm which when given the string of $N$ ones, it outputs an $N$-node $K$-Ramsey graph. (I know this is "stronger" than what you would like, but you should still check these things out for fun.)
Before the above paper, the best known explicit construction was by Frankl and Wilson, who showed that there are $2^n$ node graph that are about $2^{\Omega(\sqrt{n})}$-Ramsey. Noga Alon had an alternative construction but I think it only matched Frankl and Wilson. See the above paper for more details.
All these constructions are very neat and use radically different methods from simple counting arguments, so I hope you enjoy them. You may find that the problem of finding a succinct/effective description of a family of lower bound graphs is indeed interesting.