IP-star set (${\mathrm{IP}}^{\ast }$)

An IP-star set meets every IP set. Equivalently, its complement is not an IP set: a disjoint IP set is precisely an IP subset of the complement. This is a condition on all infinite finite-sums sets, not merely on a single generating sequence.

The IP-star sets form a proper filter on a set over the positive integers. Indeed, sets that are not IP sets are closed under taking subsets. They are also closed under finite unions, because the partition regularity of IP sets would otherwise put an IP set in one of the union's constituent sets. Taking complements gives upward closure and closure under finite intersections for the IP-star sets. The full set belongs to this filter and the empty set does not.