Interior empty-column insertion

Determine whether there exist forbidden 0-1 matrices P and P′, with sat(n, P)=O(1) and sat(n, P′)=Θ(n), such that P′ is obtained from P by inserting an empty interior column.

Background

The paper establishes that adding an empty column before the first column can change bounded saturation to linear saturation. It also proves that, with one dimension fixed, inserting empty columns preserves boundedness under the stated hypotheses. The corresponding question for an empty column inserted strictly between the first and last columns remains unresolved.

References

It remains unknown whether there exist patterns P , P ′ with sat(n, P ) = O(1) and sat(n, P ′) = Θ(n), for which P ′ is derived from P by adding an empty interior column, i.e., between the first and last columns of P .

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Section 4, immediately before Proposition 4.5 (p. 11)