Given a $k>0$ and a graph $G(V,E)$ with known independence number $\sqrt{|V|}\leq\alpha(G)\leq\alpha\sqrt{|V|}$ and chromatic number $\frac1\beta\sqrt{|V|}\leq\chi(G)\leq\sqrt{|V|}$ for some fixed $\alpha,\beta\geq2$ and given an optimum vertex coloring is it $\mathsf{NP}$-complete to decide if clique number $\omega(G)>k$? 
Deciding $\omega(G)>k$ when $\alpha(G)$ and $\chi(G)$ have bounds and are known
graph algorithms;np complete;graph colouring;clique;independent set