My cat got into a fight and his fingernail started to get infected. Now it is swolen? How do I reduce the swelling?

Neosporin or other antibiotic ointment, a band-aide if you truly feel its necessary, ice (if anything it'll numb it so it doesn't hurt as much, but again probably not really necessary), and time. If it gets or is HUGE as in twice the size of your other fingers then go see a doctor. Otherwise I think you'll survive.

If it's your cat, probably antibiotic ointment is safe but check with a vet if you think it's too swollen, again you could try icing it as well. It'll probably take a little bit of time because of the anatomy of the nail in cats.

Which is it? You cut YOUR fingernail too short and now it's swollen

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Your cat got into a fight, his fingernail is infected and swollen?

You have posted both versions.

If it's your cat, then make sure it's clean and there's no cut to get infected. If there is a cut, clean it a few times a day and keep an eye on it. Since their claws retract, it's easy for cats to get infections around them.

As long as it's kept clean (regardless if it's the cat or you) the swelling will go down on it's own. It'll just take time and the effort to keep it clean.