The hair itself doesn't trigger allergies most times. Saliva and dander (skin flakes) often are the culprit.
Consider renting a steam cleaner with a hand held wand, it will clean up the allergens and get a fair amount of the hair, too.
The other alternative is to get a dog. I'm allergic to them and the longer I am around them the less the symptoms I have. Eventually I build up an immunity.