Cutting is a very well understood process - but sadly also very dependent on some variables that are not mentioned in your question, namely the material being cut, the speed with which the sword is supposed to pass through the material, and the speed with which the chain is moving.
I am going to assume that the variables are to be set by the answerer: Everything, 30m/s, 12000m/s (yes, that's alot)
Sword-Speed is negligible to chain speed, therefore subsequent teeth will not encounter much additional material. This is both good and bad: Good in that there is not much mass to be displaced by each tooth, bad in that the edge takes the brunt of the force, and will thus blunt fast. It means you'll not have a circular chain, but rather something that is unspooled from a source and used only once. The teeth can be extremely small, the leading edge practically vertical, the edge, then a taper;The teeth might be so small the edge looks flat from the enemies perspective (and moving will appear so for sure)
12000m/s is the speed of sound in diamond, so there should not be many materials able to withstand this sword. On the edge. As this presumes a one-off chain, the source needs be in the grip, and the chain is left to unspool freely. To cut through 3 m of material will take you 1/10 th of a second, meaning 1200m of chain get unspooled. A sword grip being about 0.0001 m3 large, this comes out to about 10^-8 m2, or, for instance, 1 mm broad and 10 um thick (making the teeth be at the very limit of haptic sensibility (unmoving)). The blade cannot be thicker than the chain is broad, but thinner is better here anyways, as we do not want to move too much material.
How to get the chain up to speed is left to the reader..