Second Order Zarankiewicz Number
Abstract
We introduce the \emph{second order Zarankiewicz number} z2(m,n) for irreducible doubly simple biquadratic forms with |E1|=z(m,n), together with the intermediate recursive-line and signed parameters zRL(m,n) and zSL(m,n). They satisfy the unconditional hierarchy BSR(m,n) ≥ z2(m,n) ≥ zSL(m,n) ≥ zRL(m,n) ≥ zwL(m,n) ≥ z(m,n), where zRL is defined by the strengthened recursive rectangle criterion (RW3+) together with the conditions (S) and C4-freeness of G1, and zSL is defined by the signed criterion (RW3±). We show that (RW3+) is sound and strictly weaker than the literal weak cross-cell test (W3) on the weak-admissible class, and that (RW3±) is likewise sound. At the smallest four-column cases we obtain z2(5,4)=zSL(5,4)=zRL(5,4)=13>12=zwL(5,4), and z2(6,4)=zSL(6,4)=zRL(6,4)=16>14=zwL(6,4). In three columns this yields z2(m,3)=zRL(m,3)=2m for all m≥ 3, with strict separation from zwL(m,3) for every m≥10, and exact gap ⌊(m−3)/3⌋ for m≥16. Further finite computations give zRL(5,5)=17, zRL(7,4)=19, and zRL(7,7)≥32>28=zwL(7,7). Along N=2p with p an odd prime, we obtain the cubic asymptotic separation z2 (binomN2,N)−zwL (binomN2,N)≥ (1/16−o(1))N3. For the exceptional incidence case p=3 we prove zSL(15,6)=z2(15,6)=60. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all m≥15. The conjectural equality z2=zRL is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.
Related
- The Second-Order Augmented Zarankiewicz Number
- Asymptotic equivalence and exact values for second-order Zarankiewicz numbers
- Overtake and the recursive-line equality for second-order Zarankiewicz numbers
- Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families
- Exact Values, Extremal Classifications, and Sum-of-Squares Reductions for Second-Order Zarankiewicz Numbers
BibTeX
@misc{lofberg2026second,
title = {{Second Order Zarankiewicz Number}},
author = {Johan Löfberg and Liqun Qi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30555},
eprint = {2608.30555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30555},
}
RIS
TY - UNPB
TI - Second Order Zarankiewicz Number
AU - Johan Löfberg
AU - Liqun Qi
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2608.30555
UR - https://arxiv.org/abs/2608.30555
AB - We introduce the \emph{second order Zarankiewicz number} z_2(m,n) for irreducible doubly simple biquadratic forms with |E_1|=z(m,n), together with the intermediate recursive-line and signed parameters z_RL(m,n) and z_SL(m,n). They satisfy the unconditional hierarchy BSR(m,n) ≥ z_2(m,n) ≥ z_SL(m,n) ≥ z_RL(m,n) ≥ z_wL(m,n) ≥ z(m,n), where z_RL is defined by the strengthened recursive rectangle criterion (RW3^+) together with the conditions (S) and C_4-freeness of G_1, and z_SL is defined by the signed criterion (RW3^±). We show that (RW3^+) is sound and strictly weaker than the literal weak cross-cell test (W3) on the weak-admissible class, and that (RW3^±) is likewise sound. At the smallest four-column cases we obtain z_2(5,4)=z_SL(5,4)=z_RL(5,4)=13>12=z_wL(5,4), and z_2(6,4)=z_SL(6,4)=z_RL(6,4)=16>14=z_wL(6,4). In three columns this yields z_2(m,3)=z_RL(m,3)=2m for all m≥ 3, with strict separation from z_wL(m,3) for every m≥10, and exact gap ⌊(m−3)/3⌋ for m≥16. Further finite computations give z_RL(5,5)=17, z_RL(7,4)=19, and z_RL(7,7)≥32>28=z_wL(7,7). Along N=2p with p an odd prime, we obtain the cubic asymptotic separation z_2 (binomN2,N)−z_wL (binomN2,N)≥ (1/16−o(1))N^3. For the exceptional incidence case p=3 we prove z_SL(15,6)=z_2(15,6)=60. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all m≥15. The conjectural equality z_2=z_RL is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.
ER -