VC-dimension-only bound for the coindex of total sign complexes

Determine whether the coindex of the sign complex associated with a total sign matrix can be bounded solely as a function of the matrix’s VC dimension, independently of the number of columns.

Background

For a total sign matrix with N columns and VC dimension d, the paper proves the upper bound ind |S(A)| = O(d log2 N), and hence the same bound for coind |S(A)|. The remaining dependence on N is not known to be necessary or removable.

The cited open problem asks whether coind |S(A)| is bounded by a function of VC(A) alone. Resolving it would substantially strengthen the paper’s VC-dimension-based index estimates for total sign matrices and their associated sign complexes.

References

Removing the dependence on N remains open.

— Superpolynomial lower bounds for vertex numbers of real projective space triangulations via a topological Figiel-Lindenstrauss-Milman theorem  (2609.10402 - Frick et al., 9 Sep 2026) in Section “Sign complexes,” paragraph following Corollary 9 (discussion of Question 9 of FHV)