What are the current best lower bounds for off-diagonal Ramsey numbers R(k,l) with l of order unity and asking for asymptotic behavior for large k, such as R(k,4), R(k,5), and so on? (please include any log factors, too!) Other than the more complicated arguments of Kim for R(k,3), are all the other best lower bounds from the Lovasz local lemma?

What are the current best lower bounds for off-diagonal Ramsey numbers $R(k,l)$ with $l$ of order unity and asking for asymptotic behavior for large $k$, such as $R(k,4)$, $R(k,5)$, and so on? (please include any log factors, too!) Other than the more complicated arguments of Kim for $R(k,3)$, are all the other best lower bounds from the Lovasz local lemma?