Thanks for your input! ..... If carpet is not desirable (because of allergies), what other options are there to minimize the noise of children's running (2 and 4)? They are not out of control, running wildly all over the place. Most of the time it is just running to another part of the room. As an example, I'll be at one end of the room and they are at the other (the room is 10 paces across) and they want to come to me. Sometimes they run to go into the other room or run to sit on the potty. Things like that. It's not stomping-running. It's more of a pitter-patter kind of run.
The neighbors bothered by this noise live next door to us (on the same floor). It's the grandmother that complains. But she also told us that the neighbors upstairs complained that we were too loud too. I highly doubt this! The upstairs family has 2 kids the same age as ours (slightly younger) and I hear their kids running all the time, even up to 2 am in the morning. Which, btw, is not loud at all, just enough so that I can hear it, so it doesn't even bother me at all. DH and FIL didn't even know, until I told them to listen carefully, and then they could hear it too. I think she must be lying about the upstairs family (not about my kid's running, which she can hear).
But I do wonder, if children run the way I described, are they being noisy or is the neighbor just someone with extremely sensitive hearing?
Is it reasonable to implement a "no-running" policy for under-age-5 kids if you have neighbors living next door?