I had this very problem yesterday. It may not be just an emulsion paint issue.
Off white wool carpet. Can of Emulsion went over on Friday. Gave good coverage in one coat (about the size of 2 dining tables)
She rented a machine and removed the 'paint' element, and I was called in to remove the pigment. OneStep, soft brushes in host m/c and acid rinse (did I mention she'd overwet it on Friday?) Lots of drying, then treat with Sodium Met. and rinse. Repeated as necessary. I'd say about 95% successful. Now just a pinkish hue to the area rather than a deep blush pink all over. Customers happy enough. £500 insurance excess (don't ask) on £400 worth of carpet only 7 months old.
Safe and happy cleaning:)
Ken