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 $F(p-q_0,p,p,p-q_1)$, 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.