TITLE: Can one make Erdős's Ramsey lower bound explicit?
QUESTION [13 upvotes]: Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?
That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some $c>0$, with no monochromatic complete subgraph of size $k$?

REPLY [14 votes]: Finding explicit constructions for Ramsey graphs is a central problem in extremal combinatorics. Indeed, computational complexity gives a way to formalize this problem. Asking for a graph which can be constructed in polynomial time is a fairly good definition although sometimes the definition is taken as having a log-space construction.
Until quite recently the best method for explicit construction was based on extremal combinatorics. The vertices of the graphs were certain sets (say k-subset of an n element sets) and the edges represented pairs of sets with presecibed intersection. The best result was by Frankl and Wilson and it gives a graph with n vertices whose edges are colored by 2 colors with no monochromatic clique of size $\exp (\sqrt{(\log n))}$. (I think this translates to $k^{\log k}$ in the way the question was formulated here.) Using sum-products theorems Barak Rao Shaltiel and Wigderson improved the bound to $\exp (\log n^{o(1)})$.
Payley graphs are conjectured be explicit examples for the correct behavior. But proving it is much beyond reach.
Update(Nov 11, 2015):  Gil Cohen found an explicit construction with no monochromatic cliques of size $2^{(\log \log n)^L}$. An independent construction which applies also to the bipartite case was achieved by Eshan Chattopadhyay and David Zuckerman