Hugo, I applaud you on finding the solution, yet in your post, "Solution", you infer that four persons are to designated A, B, C, D and any additional persons assigned numbers. This was unclear, and your example lacked explanation in making it understood clearly. I assume your example for 5 people was meant to infer the following:

Let p{g} be the gossip known by person p such that {g} is the set of gossip elements known by him. Let {*} represent the complete set of all the gossip elements.

A{A}, B{B}, C{C}, D{D}, 1{1}

A calls 1: A{A,1}, B{B}, C{C}, D{D}, 1{A,1}

A calls B: A{A,B,1}, B{A,B,1}, C{C}, D{D}, 1{A,1}

C calls D: A{A,B,1}, B{A,B,1}, C{C,D}, D{C,D}, 1{A,1}