A question I have here is what do you mean by "explicit"?

Personally, I like the definition that a construction is explicit if it can be constructed in polynomial time (due to Alon? Wigderson??). Given that we are talking about exponentials in n here, this gets (slightly) complicated, but we'll say the controlling parameter here is $N=2^n$, the rough order of the number of vertices in a possible Ramsey graph.

One conjecture I have is that the set of Paley graphs on p vertices, where p ranges over all primes $1 \mod 4$ between $2^{(n/2)}$ and $2^{(n-1)}$ gives a lower bound on $R(n)$. This is NOT an explicit set, by my definition above. ::::grin:::::

If memory serves me, I think the best result known for your original question is in a paper of Noga Alon from a few yrs back. You may want to check his web page as well as Gasartch's survey page mentioned before.