TITLE: Best lower bound for off-diagonal Ramsey numbers
QUESTION [5 upvotes]: 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?

REPLY [8 votes]: 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.