Tightness of the optimized lower bound for general two-rowed configurations

Establish whether the optimized lower bound in Proposition 1.9 is tight for \(\operatorname{forb}(m,3,F(p-q_0,p,p,p-q_1))\) in the cases covered by the stated parameter range.

Background

Proposition 1.9 gives a general lower bound for the forbidden number of F(p−q0,p,p,p−q1)F(p-q_0,p,p,p-q_1), obtained by distributing the relevant non-edges between the first and last portions of the row order and optimizing over the associated parameters. The paper proves that this lower bound is tight in many cases through Theorem 1.6, but does not establish tightness in general. The authors explicitly identify the possibility that a more involved version of their proof could prove tightness in several additional cases.

References

We suspect that a more involved version of this proof could potentially show that after optimization, the general lower bound for \operatorname{forb}(m, 3, F (p − q_0, p, p, p − q_1)) given by Proposition 1.9 is also tight in several cases.

— Multivalued forbidden numbers of two-rowed configurations -- the missing cases  (2502.04741 - Peaslee et al., 7 Feb 2025) in Section 1, immediately following Proposition 1.9

A structural classification of the extremal augmented graphs remains open. The number of holes is forced by Theorem~\ref{thm:main}, but their possible positions, the allowable singleton-row distributions, and the nonisomorphic two-edge patterns are not classified. It is natural to ask whether every sufficiently large extremal graph has a bounded-size portion that can be reduced by a local rewiring operation, or whether genuinely different infinite families occur.

— Recursive-Line Zarankiewicz Numbers with Four Columns  (2609.11093 - Chen et al., 10 Sep 2026) in Section 5.2, “Structure, certificates, and higher widths”