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A347473
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Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 3 X 3 zero submatrix.
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10
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0, 2, 4, 9, 15, 21, 31, 39, 51, 63, 76, 90, 104, 127
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OFFSET
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3,2
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COMMENTS
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Related to Zarankiewicz's problem k_3(n) (cf. A001198 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 3 X 3 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 = 3, there must not be any nonzero entry in an n X n = 3 X 3 matrix, if one wants a 3 X 3 zero submatrix, whence a(3) = 0.
For n = 4, having at most 2 nonzero entries in the n X n matrix guarantees that there is a 3 X 3 zero submatrix (delete, e.g., the row which has the first nonzero entry, then the column with the remaining nonzero entry, if any), but if one allows 3 nonzero entries and they are placed on the diagonal, then there is no 3 X 3 zero submatrix. Hence, a(4) = 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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