  The characterization of the set of quantum correlations is a problem of
fundamental importance in quantum information. The question whether every
proper (tight) Bell inequality is violated in Quantum theory is an intriguing
one in this regard. Here, we make significant progress in answering this
question, by showing that every tight Bell inequality is violated by 'Almost
Quantum' correlations, a semi-definite programming relaxation of the set of
quantum correlations. As a consequence, we show that many (classes of) Bell
inequalities including two-party correlation Bell inequalities and
multi-outcome non-local computation games, that do not admit quantum
violations, are not facets of the classical Bell polytope. To do this, we make
use of the intriguing connections between Bell correlations and the
graph-theoretic Lov\'{a}sz-theta set, discovered by Cabello-Severini-Winter
(CSW). We also exploit connections between the cut polytope of graph theory and
the classical correlation Bell polytope, to show that correlation Bell
inequalities that define facets of the lower dimensional correlation polytope
are violated in quantum theory. The methods also enable us to derive novel
(almost) quantum Bell inequalities, which may be of independent interest for
self-testing applications.
