Could it be dust mites? Mold? Just asking, because if the previous owners said there were no cats (of course they could be lying), then it may actually be something else, and it would be a real pain if you were going after the wrong problem... just food for thought.
Dust mites are a real pain in ____@____.com is everywhere, and you'll never eliminate all of it, so the odds are already stacked against you. But YES, there are some things you can do, starting with where they are most prevalent, and going from there. You can do research online, and there are many websites out there that will give you a run down of the most effective ways to limit exposure to dust mites.
It's not actually the mite itself that causes the allergy, but their droppings, and their body particles when they die that become airborne, and we breathe that in. Nice, I know. Dust mites prefer dark, warm, moist places, which is why they are so prevalent in our bedding and mattresses. It's the perfect living/breeding ground for them. They recommend encasing your mattress and pillows in dust mite proof cases, so no particles can escape into the air from any mites that are in, or that get into those places. They recommend taking down any curtains. Wash all bedding, pillows, blankets, sheets, etc. at least once per week in hot, and dry in hot. Pull up any carpeting, if possible. This may be one of your culprits, if it is older carpet. You may have a lot of dust mite droppings and body particles circulating from walking on older carpets. They recommend taking down mini blinds and replacing with roll blinds, as mini blinds collect dust. Get a good vacuum with a hepa filter. Change your air filters once per month and use the anti-allergy type. And dust with a damp cloth, to keep particles from becoming airborne. That's the gist of it. It can be overwhelming. We just found out last week that my son has a ton of environmental allergies, one of which is dust mites. So we are just delving in to this area ourselves, which is why this info is so fresh in my mind.