Why exactly Eames decides to begin flirting wildly with Ariadne, no one's exactly sure.
It only ever happens in front of Arthur, where Eames will compliment the young architect on everything from her models to the color of her 'gorgeous eyes.' What begin as just compliments that could easily be interpreted as a level of politeness that belongs to other people naturally (people other than Eames, who Arthur considers to have the manners of a knuckle dragging caveman) only escalate into heavy innuendo and then even further into the realm of blatant come ons.
The innuendos are particularly emphasized by the gravel in his voice and the faux-charm his accent lends him without him even needing to try for the effect he has.
By the time it crosses the line into brushes of his fingers against her cheek in the excuse to brush back a wayward strand of her hair, or an especially close standing of their bodies as he leans in with her to look at a model, Arthur decides that enough is enough.
Without looking away from a book of artwork by Giovanni Battista, he says, "Will you two either fuck or stop talking, already? Some of us are attempting to focus."
It's hard to tell whether Ariadne or Eames is more put out by his reaction.