There are many results in number theory, where the existence of some $B \subseteq \mathbb{N}$ with certain properties is proved by a probabilistic argument employing "random sets". One such example would be the result of Erdos and Renyi, where they proved the existence of a "think" $B_2[g]$ sequence.

My question is are there certain known properties, so that if $A \subseteq \mathbb{N}$ satisfies them, then $A$ is considered to behave like a random set? or what properties would make $A$ considered to be close to being random? Thank you!