Plain language

What this result means

This result matters because every claimed number has to be carried by an explicit matrix. A larger lower bound is not a guess: it is a concrete 0/1 object whose forbidden 3 by 3 pattern can be checked directly.

  • The largest gain is z(12,21), where the lower bound moves from 116 to 126.
  • The headline small witness is z(13,17;3,3) >= 110.
  • Two values are pinned exactly because the new lower bound reaches the known upper bound: z(10,20)=102 and z(11,18)=101.

Visual notes

How to read the result

Heatmap of Zarankiewicz lower-bound improvements across m and n.
Frontier heatmapTop number is the known upper bound; bottom number is the Numaro lower bound. White cells are exact values.
Three black-and-white Zarankiewicz witness matrices loaded from saved JSON files.
Actual witness matricesThree stored 0/1 matrices: the two exact cells and the headline z(13,17) witness.
Horizontal bar chart showing the largest Zarankiewicz lower-bound gains.
Largest movesThe biggest frontier move is z(12,21), from 116 to 126 ones.

Result table

Thirty-one Zarankiewicz lower bounds raised, with two exact values pinned.

CellBaselineNumaroDeltaNote
z(12,21;3,3)116126+10largest gain
z(12,23;3,3)125134+9also follows past a monotone anchor
z(16,19;3,3)132141+9explicit witness
z(14,23;3,3)138146+8explicit witness
z(10,20;3,3)99102+3exact; reaches known upper bound
z(11,18;3,3)97101+4exact; reaches known upper bound

Method

How it was found

The campaign used two levers: monotonicity of the table and exact search for K3,3-free matrices. Each new witness was then propagated through the grid where adding a zero row or column preserves validity.

  • Read the known lower and upper bounds cell by cell.
  • Applied monotonicity: increasing m or n cannot make the best lower bound smaller.
  • Solved selected cells with an OR-Tools CP-SAT model for K3,3-free matrices.
  • Verified every saved matrix and excluded z(12,17), where the saved witness did not beat the baseline.

Verification

How it was checked

The checker counts 1s and tests every triple of rows. If any three rows share three columns, the matrix contains a forbidden 3 by 3 all-ones block and the witness fails. The 31 reported matrices pass this direct check.

Scope

What is not being claimed

Except for z(10,20)=102 and z(11,18)=101, these are improved lower bounds, not exact values. The baseline numbers are the previously published lower and upper bounds used for comparison.

References

Baseline sources

Citation

How to cite

Numaro Autoresearch Team. "New lower bounds for Zarankiewicz numbers." Numaro Research Report NUMARO-2026-008, 2026.

@techreport{numaro2026Zarankiewicz2026,
  title = {New lower bounds for Zarankiewicz numbers},
  author = {Numaro Autoresearch Team},
  institution = {Numaro},
  number = {NUMARO-2026-008},
  year = {2026},
  url = {https://numaro.tech/research/zarankiewicz-2026/}
}