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

A note on Hedetniemi's conjecture, Stahl's conjecture and the Poljak-R\"odl function