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We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.

Blokhuis showed that Paley graphs with square order have the Erdős-Ko-Rado (EKR) property in the sense that all maximum cliques are canonical. In our previous work, we extended the EKR property of Paley graphs to certain Peisert graphs and generalized Peisert graphs. In this paper, we propose a conjecture which generalizes the EKR property of Paley graphs, and can be viewed as an analogue of Chvátal's Conjecture for families of set systems. As a partial progress, we prove that maximum cliques in pseudo-Paley graphs have a rigid structure.