For a prime power "q congruent to 1 modulo 4", a Paley graph is the graph
with vertex set the finite field on q elements, "F_q", and an edge between
two of its vertices if and only if their difference is a non-zero square
in "F_q".
A key feature of Paley graphs is that they mimic the typical behaviour of
a random graph "G(q,1/2)", that is the graph on q vertices where each
possible edge is present or absent equiprobably, independent of all other
edges.
A particular reason for interest in Paley graphs is the long-standing
suspicion that some of them may provide better lower bounds for diagonal
Ramsey numbers than the current best graphs "G(q,1/2)".
The aim of this talk will be the discussion of Paley graphs, their
connection with Ramsey numbers and the theory of Extremal Ramsey graphs.