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.