I'm not really understanding the question. I'm imagining a counterexample where people come in two at a time, meet, and leave, with people coming in enough times so that every 3-subset of the set of all visitors contains at least one pair of people that have met. For example, with 5 people,

{1,2}, {1,4}, {1,3}, {2,4}, {2,5}, {3,5}.

What am I doing wrong?