I am rehabbing a house that does not currently have a laundry tub in the basement. I have to cut into the cast iron pipe and add a clean out anyway, and would like to add a laundry tub. The only problem is that the house is two and a half stories tall and I have no way of getting a vent pipe fed up high enough to get it connected back into the main stack above where the second floor drains come into the main stack. My question is, I think I have heard that if the trap from the tub is within a minimum number of inches from the main stack I don't need to run a separate vent stack. I can place the tub right next to the main stack so the drain would only run about foot till it's in the main stack. Any thoughts?

Place your laundry sink as close to the main stack as possible. Use 1 1/2" drain pipe. Slope 1/4" per foot. If it less than 4' from the stack vent than I would not worry about vent at all. If more than 4' than install AAV vent behind the trap.

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I think I have heard that if the trap from the tub is within a minimum number of inches from the main stack I don't need to run a separate vent stack. I can place the tub right next to the main stack so the drain would only run about foot till it's in the main stack. Any thoughts?

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Not quite so. Milo erred in his answer. You may connect to a vent stack( dry vent) but all the stacks coming into your basement are stack vents,( vents with other fixtures draining into it) and you can not discharge a major fixture past a unvented minor one.
But Milo did leave a ray of hope when he suggested installing a AAV for a vent,(see image) You may connect to the stack using va AAV and be vented. Good luck, Tom