Exact forbidden number for the multivalued identity configuration

Determine the exact value of \(\operatorname{forb}(m,3,p\cdot I_2)\) for general \(p\), where \(p\cdot I_2\) is the configuration formed by concatenating \(p\) copies of the two-row identity matrix.

Background

The paper compares the forbidden numbers of the configurations p⋅K2p\cdot K_2 and p⋅I2p\cdot I_2 for 3-matrices. It establishes the exact value of forb⁡(m,3,p⋅K2)\operatorname{forb}(m,3,p\cdot K_2) in the relevant range and proves that the difference between the two forbidden numbers is eventually independent of mm, with asymptotic size 12p(log⁡2p)2\tfrac{1}{2}p(\log_2 p)^2. However, the argument relies on bounds for forb⁡(m,3,p⋅I2)\operatorname{forb}(m,3,p\cdot I_2), whose exact value is not determined for general pp.

References

By [DS21], if 2{m−2} ≥ p − 1, then \operatorname{forb}(m, 3, p * K_2) = 2m + m2{m−1} + (p − 1)\binom{m}{2}, but the exact value of \operatorname{forb}(m, 3, p * I_2) is not known for general p.

— Multivalued forbidden numbers of two-rowed configurations -- the missing cases  (2502.04741 - Peaslee et al., 7 Feb 2025) in Section 2.1, “Asymptotics of \(\operatorname{forb}(m,3,p\cdot K_2)-\operatorname{forb}(m,3,p\cdot I_2)\)”