My own au I thought up. Basically, Error and Ink are much MUCH older than any other sans, they even make the likes of nightmare and dream seem young. Because of this, they often go through moments of being friends, being sworn enemies, fighting to the death, or seeking comfort from eachother. They often argue and fight, and when they become lonely from the fact they can't completely connect with other people as they haven't lived as long as them, they'll seek eachother out and sometimes even make a temporary truce. It's all random though.
However, one such time when they are battling, PaperJam, or PJ is born from their magic. At first they would be taught individually by each, and be in the middle of their fighting. Then one day Gradient was born.
Error and Ink never realized the pain they were causing to the two until PJ snapped and yelled at them. This made the two talk it over and decide that on the weeks of each of their children's birthdays, a truce would happen. And that they must never drag the two into the battles or force them to pick a side.
This is my own au, as I have yet to see someone do an au like this. Not necessarily ErrorInk, but I'll put it in the tags anyways as they do have moments, I guess.