The widest tire that is recommended to put on a rim is where the rim width is 80% of the the tread width. So, for a 6" wide rim, the max tread width should be about 7.5". Any wider, the sidewalls will begin to loose strength inducing less strength to remain "standing" when side or lateral loads are applied. The wider you go past this, the less strength you have.
The most narrow tire should have a rim width of 90% of the tread width. For the 6" rim, this would be about 6.75".
Now befor anybody has a cow or
?, please remember you can go wider or smaller, but this is the recommendation from most tire manufactureres with regards to maximum handling capabilities for a tire. 2nd but......always check with the tire manufacturer whose brand you are about to buy for the recommended tire size versus rim, you would be surprised at the differences between mfg's with the exact same tire size versus their design specs!