Question for people knowledgable about Ramsey theory - with current computing tech, will we ever be able to find the exact value of R(5,5) within our lifetimes? I've read the famous Erdos quote about R(5,5) vs R(6,6) and he seemed to think that it was at least theoretically possible if the whole human race had to find it quickly or face destruction.
As others have said, brute force is just plain impossible. But we can limit the search space significantly by proving sub-results about the structures of the possible graphs. Somewhat trivial example to give the flavor: let's state the problem as "find the smallest number of vertices needed such that any graph must have either a 5-connected component (a "pentagram") or 5 independent vertices". We know that the number is at least 43. So if we are looking for a counterexample, a graph on 43 vertices that does not have either of these two subgraphs, what can we say about the possible number of edges that each vertex must have? (the degree, for graph theorists). We can immediately say that if we have 5 or more vertices with no edges (degree 0) then we have our independent set already, so any counterexample can have at most 4 vertices of degree 0.
By the way, my favorite Explain-like-I'm-5 version of Ramsey's theorem is "Total disorder is impossible": If a structure is large enough, there must be some substructure that is ordered.