Improved Lower Bounds on the Classical Ramsey Numbers R(4,22) and R(4,25)

Circulant graphs have been used to effectively establish lower bounds on many classical Ramsey numbers. Here, we construct circulant graphs of prime order that sharpen the best published lower bounds on two Ramsey numbers. Generalizing previous work in which quadratic and cubic residues were used to construct circulant graphs for the same purpose, we report on the use of quartic and higher-order residues.