The calamity all started at the beginning of L.A.'s game against Chicago ... when one witness tells us a Bears fan got into a verbal argument with several Rams fans for cheering the action on the field.

The witness says things got so aggressive in the seats, security at SoFi Stadium had to step in and request people swap chairs in order to keep the peace.

We're told after the game, though, some of the men ran into each other inside the concourses at the newly built arena -- and it didn't take long for the fists to start flying.

In video of the battle, you can see multiple scraps broke out ... including one where a man in an Akiem Hicks jersey was pummeled by punches and kicks.

In fact, the fights were so violent ... you can see blood actually ended up smeared all over the man's Hicks jersey.

We're told security was eventually able to break things up ... ultimately putting one of the combatants in handcuffs -- though it's unclear if any actual arrests were made.