Determine the linear-order asymptotic deficit of the maximum irredundant cover size

Determine the finer asymptotic behavior of the deficit \(\frac12m^2-F(m)\) for the maximum number \(F(m)\) of \(2\times2\) cards in an irredundant cover of an \(m\times m\) grid, specifically whether there exists a constant \(c>0\) such that \(F(m)=\frac12m^2-cm+O(1)\), or whether the linear coefficient depends on the residue class of \(m\) modulo \(4\).

Background

The paper defines F(m)F(m) as the maximum cardinality of an irredundant cover of an m×mm\times m grid by axis-aligned 2×22\times2 cards. Its uniform upper bound and period-four staircase lower-bound construction establish only that F(m)=12m2−Θ(m)F(m)=\frac12m^2-\Theta(m), equivalently that the deficit from the leading term 12m2\frac12m^2 is linear in mm.

The authors explicitly leave unresolved the coefficient and finer structure of this linear deficit. They ask whether a single constant governs the correction term up to bounded error, or whether separate coefficients are required for the four residue classes modulo $4$, as suggested by the period-four staircase construction and boundary repairs.

References

Main open problem. The preceding bounds show that the deficit \frac12m2-F(m) has linear order. The remaining problem is to determine its finer asymptotic behaviour. In particular, decide whether there is a constant c>0 such that F(m)=\frac12m2-cm+O(1), or, more generally, whether there are constants c_0,c_1,c_2,c_3>0 such that F(m)=\frac12m2-c_r m+O(1)\qquad(m\equiv r\pmod4).

— Irredundant Covers of Square Grids by 2 x 2 Cards: A Defect Framework  (2609.10063 - Eruysal et al., 9 Sep 2026) in Main open problem subsection following Theorem (after Section 4, Lower bounds)