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Blokhuis showed that all maximum cliques in Paley graphs of square order havea subfield structure. Recently it is shown that in Peisert-type graphs, somemaximum cliques do not have a subfield structure, and yet all maximum cliqueshave a subspace structure. In this paper, we investigate the existence of aclique of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs oforder $q$ from unions of semi-primitive cyclotomic classes. We show that such aclique must have an equal contribution from each cyclotomic class and that mostpseudo-Paley graphs do not admit such cliques, suggesting that the Delsartebound $\sqrt{q}$ on the clique number can be improved in general. We also provethat generalized Peisert graphs are not isomorphic to Paley graphs or Peisertgraphs, confirming a conjecture of Mullin.

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes