We present a new polynomial-time algorithm for finding maximal
cliques in graphs. As a corollary, we obtain new bounds on the
famous Ramsey numbers in terms of the maximum and minimum vertex
degrees of the corresponding Ramsey graphs. The algorithm finds a
maximum clique in all known examples of graphs. In view of the
importance of the P versus NP question, we ask if there exists a
graph for which the algorithm cannot find a maximum clique. The
algorithm is demonstrated by finding maximum cliques for several
famous graphs, including two large benchmark graphs with hidden
maximum cliques. We implement the algorithm in C++ and provide a
demonstration program for Microsoft Windows.
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