Square-difference-free set (https://en.wikipedia.org/wiki/Square-difference-free_set), now a Good Article on Wikipedia.
As the name suggests, these are sets of integers no two of which differ by a square. My favorite is the losing positions in subtract-a-square (https://en.wikipedia.org/wiki/Subtract_a_square), where each move removes a square number of coins from a pile of coins, winning by taking the last coin.
This general class of sets and the subtract-a-square set have \(o(n)\) elements up to \(n\), but their maximum density remains unknown.
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