What are dense Sidon subsets of {1,2,…,n} like?

The short answer if you don’t feel like reading a post with some actual mathematics in it is that I don’t know.

Now for the longer answer. A subset of is called a Sidon set if the only solutions of the equation with are the trivial ones with and or and . Since the number of pairs with is and whenever , it is trivial that if is a Sidon set, then , which gives an upper bound for of around .