MathDB

Löfberg–Qi conjecture

For all positive integers m,nm,n, the second-order Zarankiewicz number equals the recursive-line quantity: z2(m,n)=zRL(m,n)z_2(m,n)=z_{RL}(m,n).

References

Primary source

arXiv

Additional references

extremal-combinatoricspolynomial-optimizationsum-of-squares-methods
Posted · edited

Progress summary

Refreshed
Claimed progress

New papers establish the conjectured equality in several finite cases and claim it for a broad four-column range, but the all-parameter conjecture remains open.

The Löfberg–Qi conjecture asks whether the second-order quantity always equals the recursive-line quantity, namely z2(m,n)=zRL(m,n)z_2(m,n)=z_{RL}(m,n). Löfberg and Qi introduced these quantities in a paper dated August 31, 2026; the general equality is not proved there.

Recent September 2026 developments

  • Löfberg and Qi prove zRL(5,4)=13>12=zwL(5,4)z_{RL}(5,4)=13>12=z_{wL}(5,4) and give finite evidence for the conjecture.
  • Chen and Chen claim an eventual four-column formula and state that the same formula holds for z2(m,4)z_2(m,4), implying equality with the recursive-line quantity for m≥15m\ge15; their finite analysis leaves zR(14,4)z_R(14,4) unresolved.
  • Xu and Yan report exact values z2(4,4)=10z_2(4,4)=10, z2(7,4)=19z_2(7,4)=19, z2(8,4)=21z_2(8,4)=21, and z2(5,5)=17z_2(5,5)=17, with exhaustive orbit-level checks and agreement with recursive bounds.

Current status (as of September 2026): several cases and a claimed four-column range support z2(m,n)=zRL(m,n)z_2(m,n)=z_{RL}(m,n), but the all-parameter conjecture remains open and the claims are unverified.

Sources

Solutions 0

No solutions have been posted yet.