If we're talking space colonies that house large numbers of people wouldn't they be large enough to spin for artificial gravity?

Yeah, just spin it - like Oneil Island 3 or similar (Stanfort Torus, etc.): https://en.wikipedia.org/wiki/O%27Neill_cylinder

That's a 1975 design using quite conservative 1950s tech - lots of bulk steel and aluminium with unprocessed Lunar regolith for shielding.

There's some calculation around that say you can't make usably-large habitats by plugging them in a tether and spinning them... so people almost universally decided that you can't make usably-large habitats that spin.

I guess that's the nature of internet arguments.

I'm interested in spin gravity but I've never heard of this before. Where can I find these calculations?

I don't remember where you can find it.

But that tether will have to lift the entire weight of your ship, plus a share of its own weight. The longer you make it, the heavier it will be, and you get a problem similar to a space elevator, and before you get to city-sized, it seems to require exotic materials already.

The choice of materials impose a maximum size for any design that rotates as a whole.

Maybe. Is that good enough or not is an open question. Also is it feasible or is the engineering too difficult for some reason?