  Contemporary versions of Bell's argument against local hidden variable (LHV)
theories are based on the Clauser Horne Shimony and Holt (CHSH) inequality, and
various attempts to generalize it. The amount of violation of these
inequalities cannot exceed the bound set by the Grothendieck constants.
However, if we go back to the original derivation by Bell, and use the perfect
anti-correlation embodied in the singlet spin state, we can go beyond these
bounds. In this paper we derive two-particle Bell inequalities for traceless
two-outcome observables, whose violation in the singlet spin state go beyond
the Grothendieck constants both for the two and three dimensional cases.
Moreover, creating a higher dimensional analog of perfect correlations, and
applying a recent result of Alon and his associates (Invent. Math. 163 499
(2006)) we prove that there are two-particle Bell inequalities for traceless
two-outcome observables whose violation increases to infinity as the dimension
and number of measurements grow. Technically these result are possible because
perfect correlations (or anti-correlations) allow us to transport the indices
of the inequality from the edges of a bipartite graph to those of the complete
graph. Finally, it is shown how to apply these results to mixed Werner states,
provided that the noise does not exceed 20%.
