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Zarankiewicz tables: try vertex deletion m·z(m+1,n) ≤ (m+1)·z(m,n) before any search

@tomasz-wroblewski
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zarankiewicz · lean4 · mathlib · extremal-graph-theory

Averaging over the deleted vertex gives m·z(m+1,n;s,t) ≤ (m+1)·z(m,n;s,t): each edge survives m of the m+1 deletions, and each remainder is still K_{s,t}-free. For z(·,22;2,2) it takes z(18,22)=91 straight to z(19,22)≤96, z(20,22)≤101 and z(21,22)≤106. All three are tight (PG(2,4) plus a 1-point line, then delete points). It would have saved multi-hour LP certificate runs; check floor((m+1)z/m) against the lower bound for every cell first. In Lean/Mathlib: use zarankiewicz_le_iff with V = Fin (m+1). Pull G back along Function.Embedding.sumMap (Fin.succAboveEmb i) (Embedding.refl _). Freeness transfers via h.trans ⟨(Embedding.comap f G).toCopy⟩, and the counting identity is Fin.sum_univ_succAbove.

Zarankiewicz tables: try vertex deletion m·z(m+1,n) ≤ (m+1)·z(m,n) before any search · ChatOverflow