ZarankiewiczDB,

an online database of Zarankiewicz numbers.

z(26,26) = 138

Searching for matrices of weight greater than 138, all but 6 scaffolds were eliminated without search.

Scaffold: $p=5$, $q=5$, $\mathbf{m}=(4,4,4,4,4)$, $\mathbf{n}=(4,4,4,4,4)$, $m_O=0$, $n_O=0$.

This is a single box; the scaffold weight is $51$, and the maximum interior weight is at most z(5,5 | 4,4) $= 87$, giving a total of $138$.

Scaffold: $p=5$, $q=5$, $\mathbf{m}=(5,4,4,4,3)$, $\mathbf{n}=(4,4,4,4,4)$, $m_O=0$, $n_O=0$.

(665.849771189 seconds, score = 0, 4419556 recursive calls, max depth = 18)

Scaffold: $p=5$, $q=5$, $\mathbf{m}=(5,4,4,4,3)$, $\mathbf{n}=(5,4,4,4,3)$, $m_O=0$, $n_O=0$.

(3504.343245546 seconds, score = 0, 11595939 recursive calls, max depth = 18)

Scaffold: $p=5$, $q=5$, $\mathbf{m}=(5,5,4,3,3)$, $\mathbf{n}=(4,4,4,4,4)$, $m_O=0$, $n_O=0$.

(110.345962233 seconds, score = 0, 343371 recursive calls, max depth = 14)

Scaffold: $p=5$, $q=5$, $\mathbf{m}=(5,5,4,3,3)$, $\mathbf{n}=(5,4,4,4,3)$, $m_O=0$, $n_O=0$.

(10.292675708 seconds, score = 0, 29657 recursive calls, max depth = 14)

Scaffold: $p=5$, $q=5$, $\mathbf{m}=(5,5,4,3,3)$, $\mathbf{n}=(5,5,4,3,3)$, $m_O=0$, $n_O=0$.

(0.005738312 seconds, score = 0, 0 recursive calls, max depth = 0)

upper bound = 138, submitted by Andrew Kay on April 25th, 2016 (link)

By deleting 5 rows and 5 columns from the finite projective plane of order 5.

lower bound = 138, submitted on April 25th, 2016 (link)

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