  We introduce a new model for the chromatic number $\chi(G)$ based on what we
call combinatorial projection matrices, which is a special class of doubly
stochastic symmetric projection matrices. Relaxing this models yields an SDP
whose optimal value is the projection theta number $\hat{\vartheta}(G)$, which
is closely related to the Szegedy number $\vartheta^+(G)$, a variant of the
Lov\'{a}sz theta number. We characterize that in general,
$\hat{\vartheta}(G)\leq \vartheta^+(G)$, with equality if $G$ is
vertex-transitive. While this seems to imply that working with binary matrices
is a better paradigm than working with binary eigenvalues in this context, our
approach is slightly faster than computing the Szegedy number on
vertex-transitive graphs.
