Quantum sets of the multicolored-graph approach to contextuality

The Clauser-Horne-Shimony-Holt (CHSH) inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts quantum classical correlations. For example, theta body exclusivity can be associated set correlations achieved by theory. Following Cabello-Severini-Winter (CSW) approach, one may think that CHSH $\mathrm{TH}({G}_{\text{CHSH}})$ is equal inequality ${\mathcal{Q}}_{\text{CHSH}}$, but this really true? All assumptions about come from scenarios, while CSW only demands structure (NC) scenario. To deal extra related presence different players in scenario like one, colored-graph was introduced. Does it make any difference as or more general NC scenario? represented bicolored $\mathcal{G}$ simple $G$, shadow colored $\mathcal{G}$. In general, we have ${\ensuremath{\theta}}_{c}(\mathcal{G})$ subset its $\mathrm{TH}(G)$ same way Lov\'asz number, corresponds bound, $\ensuremath{\vartheta}(G)$ greater than number ${\ensuremath{\vartheta}}_{c}(\mathcal{G})$. case inequality, $\ensuremath{\vartheta}(G)={\ensuremath{\vartheta}}_{c}(\mathcal{G})$. accident also hold for corresponding sets? Is true $\mathrm{TH}(G)={\ensuremath{\theta}}_{c}(\mathcal{G})$, would mean every correlation reached theory applied could obtained at principle restrictive paper our answer such question negative. We show ${\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremath{\nsubseteq}\mathrm{TH}(G)$ therefore there cannot under restrictions.