Second-order Zarankiewicz-number conjecture for complete-graph incidence families

For every integer $n\ge 6$ , let $m=\binom{n}{2}$ and consider the incidence family of the complete graph $K_n$ , with one column for each vertex and one row for each edge. If $z_2(m,n)$ , $z_{SL}(m,n)$ , and $z_{RL}(m,n)$ denote its second-order, signed, and recursive-line Zarankiewicz numbers, then

z_2(m,n)=z_{SL}(m,n)=z_{RL}(m,n)=Z(n),

where

Z(n)=\left\lfloor\frac{n(n-1)(n+2)}{4}\right\rfloor=\begin{cases}\dfrac{n(n-1)(n+2)}{4},&n\equiv0,1\pmod 4,\\[6pt]\dfrac{n(n-1)(n+2)-2}{4},&n\equiv3\pmod 4.\end{cases}

References

Primary source

arXiv

Additional references

Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families — arXiv — Yannan Chen, Liqun Qi

design-theory extremal-combinatorics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper claims substantial progress by settling many cases, but the conjecture remains open for infinitely many orders.

The conjecture concerns exact second-order Zarankiewicz values for incidence families arising from complete graphs. Its origin and proposer are not identified in the retrieved material.

September 2026 exact-value claims

On September 22, 2026, Yannan Chen and Liqun Qi reported exact values for infinite families of even and odd orders, plus certified configurations for $n=8$ , $n=9$ , $n=12$ , and $n=13$ . The paper explicitly leaves the remaining orders, beginning with $n=16$ , as conjectural.

Current status (as of September 2026): Exact values are claimed for several infinite families and four additional orders, while the conjecture remains open from $n=16$ onward; the claims have not been independently verified in this scan.

Sources

No solutions have been posted yet.