Finite algorithm for bounded saturation

Determine whether a finite algorithm exists that, given a forbidden 0-1 matrix P, decides whether its saturation function sat(n, P) is bounded.

Background

The saturation function of every forbidden 0-1 matrix is either bounded or asymptotically linear, but a complete structural classification is not known. Fulek and Keszegh asked whether boundedness can be decided algorithmically. The paper proves a bounded-witness-size result when one matrix dimension is fixed, but the authors do not obtain a bound independent of that fixed dimension; such a bound would resolve the original algorithmic question.

References

Fulek and Keszegh asked in [10] whether a finite algorithm exists to determine whether a pattern P has bounded sat(n, P ). As they note, this is equivalent to asking whether there exists a computable function f (k, l) such that, for a k × l pattern P , there exists a witness for P of dimensions at most f (k, l) × f (k, l). We resolve a version of this question for sat(m0, n, P ).

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Section 3, immediately before Proposition 3.4 (p. 8)