Given an odd prime power q, the Paley graph of order q^2 is the graph obtained by taking elements of the finite field F_{q^2} as vertices, where two vertices x and y are connected by an edge if x-y is a square in F_{q^2}. It is not hard to check that the subfield F_q is a maximum clique in the Paley graph. Blokhuis proved that Paley graphs of square order have the Erdos-Ko-Rado (EKR) property in the sense that all maximum cliques are canonical (that is, an affine transformation of this subfield F_q). This settled affirmatively the conjecture by van Lint and MacWilliams who had asked whether every maximum clique in the Paley graph of order q^2 containing 0, 1 must be the subfield F_q. In this talk, we present a different solution to this problem using the results on the number of directions in a finite Galois plane and recent advances on character sums over finite fields. We will also present a generalization of Blokhuis' Theorem to a larger family of Cayley graphs, which we call Peisert-type graphs, and as a consequence, resolve a conjecture by Mullin on Peisert graphs. This is joint work with Kyle Yip. The part II of the talk will be given by Kyle next week.