Existential Second Order Logic Expression With Horn First Order for Maximum Clique (Decision Version)

We show that the maximum clique problem (decision version) can be expressed in existential second order (ESO) logic, where the first order part is a Horn formula in second-order quantified predicates. Without ordering, the first order part is $\Pi2$ Horn; if ordering is used, then it is universal Horn (in which case, the second order variables can be determined in polynomial time). UPDATE: Manuscript withdrawn, because results are incorrect. If phi = phi1 AND phi2, and phi is a Horn formula, it does NOT mean that both phi1 and phi_2 are Horn formulae. Furthermore, the cardinality constraint CANNOT be expressed as a universal Horn sentence in ESO (NOT even when the structure is ordered). Graedel's theorem is valid at a lower (machine) level, but probably NOT at a higher level.