Hi
Welcome to the MedHelp forum!
If there is no pain, then the swelling could just be due to overuse of arm, blocked gland, sweat dermatitis or even a slow growing abscess. Let your husband wash the area well and apply an ice pack and see if it helps. If swelling persists then consult a doctor.
Hope this helps. Please let me know if there is any thing else and do keep me posted. Take care!
Very normal, no need to worry, combination heat, humidity and wide open pores in his arm pits allowed dirt to seep around hair folicales, very important to use antipersprant not just deoderant