Raintree, I think the key word is "mingle". Generally, at parties and other social events, people move around, and talk to various people. The OP states here that Amy and Ted spent "most of the BBQ" talking with each other. Assuming the BBQ lasted several hours, this is a significant amount of time.
A half hour chat at a party? Sure, no need to mention your significant other. But to spend several hours at a social event, talking one-on-one (rather than moving about, talking with a variety of people, as one normally does at a social event) to me indicates a level of interest that MAY go beyond "having a friendly chat with a random person at a BBQ".
So to answer cross_patch's question, I'd say that if you are going to spend several hours having an in-depth conversation with an eligible member of the opposite sex, then yes, you do need to slip into the conversation that you're taken. It doesn't need to be a major production. It can simply be something like "Oh, you like art? So do I! My boyfriend and I happened to see the recent Monet exhibition at the National Gallery the other week and it was wonderful. What do you think of the Impressionists?" etc.