Thanks for asking the query,
Red ans swollen gums
around the errupting teeth can be because of few reasons first the underlying teeth is trying to errupt and it will take its own course so there is nothing to worry.
Next is due to the food lodgement leading to redness
and swelling of the overlying gums, so you need not to worry.
Maintain a good oral hygiene
Use lukewarm saline and antiseptic mouthwash gargles twice daily.
Take a balanced nutritious diet and drink plenty of water.
Hope this helps out.