### Primitive Sets with Small Gaps

A primitive set is a set $S$ of positive integers such that no pair of distinct elements $a,b\in S$ satisfies $a | b$.

A square-primitive set is a set of positive integers $S$ such that no pair of distinct elements $a, b \in S$ has ratio $a/b$ equal to an square integer.

In the following writeup, I show how to construct square-primitive sets with very short gaps $O_{\epsilon} ((\log N)^2 (\log \log N)^{\epsilon})$. This relies on dividing the poset $\mathbb{N}$ ordered by divisibility into antichains $A_m$ consisting of all numbers in $\mathbb{N}$ with exactly $m$ (not necessarily distinct) prime factors, and then picking one randomly.

Square Primitive Sets With Small Gaps