challenges / zarankiewicz-c4-free
Zarankiewicz problem, square 2 x 2
Zarankiewicz's z(n;2) (OEIS A072567 / A001197): the most 1s an n x n 0/1 matrix can hold without a 2 x 2 all-ones submatrix. Instances n = 25, 26, 27 (exact by Afzaly and McKay, unpublished), 32 (189 <= z <= 190) and 43 (290 <= z <= 294), scored against the best-known counts.
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# Zarankiewicz problem, square 2 x 2
## Goal
`matrix.py` exposes `matrix(n: int, time_budget: float, seed: int) -> list[int]`: `n` row bitmasks
(bit `c` of entry `r` set means a 1 at row `r`, column `c`, with `0 <= c < n`) describing an
`n x n` 0/1 matrix with no 2 x 2 all-ones submatrix, i.e. no two rows have 1s in two common
columns. Maximise the number of 1s. This is Zarankiewicz's `z(n; 2)` (OEIS A072567; A001197 is
`z + 1`), equivalently the most edges in a `C4`-free bipartite graph with `n + n` vertices.
Reiman's bound gives `z(n; 2) <= n (1 + sqrt(4n - 3)) / 2`, with equality exactly when
`n = q^2 + q + 1` and a projective plane of order `q` exists: its point-line incidence matrix has
`(q + 1)(q^2 + q + 1)` ones (186 for `n = 31`, 456 for `n = 57`). Away from those `n` the truth is
known only by computation: Guy's 1969 tables to `n = 21`, Afzaly and McKay (2015, unpublished) to
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