### Primitive Sets with Small Gaps

A primitive set is a set of positive integers such that no pair of distinct elements satisfies .

A square-primitive set is a set of positive integers such that no pair of distinct elements has ratio equal to an square integer.

In the following writeup, I show how to construct square-primitive sets with very short gaps . This relies on dividing the poset ordered by divisibility into antichains consisting of all numbers in with exactly (not necessarily distinct) prime factors, and then picking one randomly.