We prove a non-existence theorem for smooth AdS5 solutions with connected, compact without boundary internal space that preserve strictly 24 supersymmetries. In particular, we show that D=11 supergravity does not admit such solutions, and that all such solutions of IIB supergravity are locally isometric to the AdS_5 * S^5 maximally supersymmetric background. Furthermore, we prove that (massive) IIA supergravity also does not admit such solutions, provided that the homogeneity conjecture for massive IIA supergravity is valid. In the context of AdS/CFT these results imply that if strictly N=3 superconformal theories in 4-dimensions exist, their gravitational dual backgrounds are either singular or their internal spaces are not compact.
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