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A347474
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Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 4 X 4 zero submatrix.
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9
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OFFSET
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4,2
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COMMENTS
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Related to Zarankiewicz's problem k_4(n) (cf. A006616 and other crossrefs) which asks the converse: how many 1's must be in an n X n {0,1}-matrix in order to guarantee the existence of an all-ones 4 X 4 submatrix. This complementarity leads to the given formula which was used to compute the given values.
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LINKS
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FORMULA
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EXAMPLE
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For n < 4, there is no solution, since there cannot be a 4 X 4 submatrix in a matrix of smaller size.
For n = 4, there must not be any nonzero entry in an n X n = 4 X 4 matrix, if one wants a 4 X 4 zero submatrix, whence a(4) = 0.
For n = 5, having at most 2 nonzero entries in the n X n matrix guarantees that there is a 4 X 4 zero submatrix (delete, e.g., the row with the first nonzero entry, then the column with the second nonzero entry, if any), but if one allows 3 nonzero entries and they are placed on the diagonal, then there is no 4 X 4 zero submatrix. Hence, a(5) = 2.
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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