We prove that $\min\{χ(G), χ(H)\} - χ(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{χ(G), χ(H)\}/χ(G\times H) \leq 1/2 + ε$ for large values of $\min\{χ(G), χ(H)\}$.
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