A001198 -id:A001198 - cp's OEIS frontend

A001197 Zarankiewicz's problem k_2(n).

4, 7, 10, 13, 17, 22, 25, 30, 35, 40, 46, 53, 57, 62, 68, 75, 82, 89, 97, 106, 109, 116, 123

Offset: 2

a(n) is the minimum number k_2(n) such that any n X n matrix having that number of nonzero entries has a 2 X 2 submatrix with only nonzero entries. - M. F. Hasler, Sep 28 2021

a(n) <= (1 + sqrt(4*n-3))*n/2 + 1. - Max Alekseyev, Apr 03 2022

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 291.
R. K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150.
Richard J. Nowakowski, Zarankiewicz's Problem, PhD Dissertation, University of Calgary, 1978, page 202.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
R. K. Guy, A many-facetted problem of Zarankiewicz, Lect. Notes Math. 110 (1969), 129-148.

Cf. A001198 (k_3), A072567, A339635, A347472, A350296.
Cf. also A006613 - A006626 (other sizes, in particular A006616 = k_4).
Main diagonal of A376167.

a(n) = A072567(n) + 1. - Rob Pratt, Aug 09 2019

a(n) = n^2 - A347472(n) = n^2 - A350296(n) + 1. - Andrew Howroyd, Dec 26 2021

Nowakowski's thesis, directed by Guy, corrected Guy's value for a(15) and supplied a(16)-a(21) entered by Don Knuth, Aug 13 2014
a(1) deleted following a suggestion from M. F. Hasler. - N. J. A. Sloane, Oct 22 2021
a(22)-a(24) from Jeremy Tan, Jan 23 2022

A350237 Minimum number of 1's in an n X n binary matrix with no zero 3 X 3 submatrix.

Original entry on oeis.org

0, 0, 1, 3, 5, 10, 16, 22, 32, 40, 52, 64, 77, 91, 105, 128

Offset: 1

Jeremy Tan, Dec 21 2021

The submatrix's rows and columns need not be contiguous, so the following matrix does not show a(4) = 1:
   ....
   .1..
   ....
   ....

			a(4) = 3 because the following 4 X 4 binary matrix with 3 1's has no zero 3 X 3 submatrix, and all such matrices with fewer 1's have at least one zero 3 X 3 submatrix:
   1...
   .1..
   ..1.
   ....

Jeremy Tan, Python program with illustration of initial terms
Jeremy Tan, Two genies and their kind of chess, Puzzling Stack Exchange, Dec 19 2021. (shows a(8) = 22)
Jeremy Tan, What's the minimum number of people required?, Mathematics Stack Exchange, Dec 20 2021.
Jeremy Tan, An attack on Zarankiewicz's problem through SAT solving, arXiv:2203.02283 [math.CO], 2022.

Column 3 of A339635.

Cf. A001198, A347473, A350304.

a(n) = A347473(n) + 1 = n^2 - A001198(n) + 1.

a(n) = n^2 - A350304(n). - Max Alekseyev, Oct 31 2022

a(12)-a(13) from Andrew Howroyd, Dec 23 2021
a(14)-a(15) from Jeremy Tan, Jan 03 2022
a(16) from Jeremy Tan, added by Max Alekseyev, Oct 31 2022

A347473 Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 3 X 3 zero submatrix.

Original entry on oeis.org

0, 2, 4, 9, 15, 21, 31, 39, 51, 63, 76, 90, 104, 127

Offset: 3

M. F. Hasler, Sep 28 2021

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.

			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.

Cf. A001198 (k_3(n)), A001197 (k_2(n)), A006613 - A006626 (other sizes of the main matrix and the submatrix).
Cf. A347472, A347474 (analog for 2 X 2 resp. 4 X 4 zero submatrix).
Cf. A339635, A350237, A350304.

a(n) = n^2 - A001198(n).
a(n) = A350237(n) - 1. - Andrew Howroyd, Dec 24 2021
a(n) = n^2 - A350304(n) - 1. - Max Alekseyev, Oct 31 2022

a(11)-a(13) from Andrew Howroyd, Dec 24 2021

a(14)-a(16) computed from A350237 by Max Alekseyev, Apr 01 2022, Oct 31 2022

A347474 Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 4 X 4 zero submatrix.

Original entry on oeis.org

0, 2, 4, 6, 12, 19, 25, 34, 43, 51

Offset: 4

M. F. Hasler, Sep 28 2021

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.

			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.

Cf. A347472, A347473 (analog for 2 X 2 resp. 3 X 3 zero submatrix).
Cf. A006616 (k_4(n)), A001198 (k_3(n)), A001197 (k_2(n)), A006613 - A006626.
Cf. A339635.

a(n) = n^2 - A006616(n).

a(n) = A339635(n,4) - 1. - Andrew Howroyd, Dec 23 2021

a(9)-a(12) from Andrew Howroyd, Dec 23 2021

a(13) computed from A006616 by Max Alekseyev, Feb 02 2024

A350304 Maximum number of 1's in an n X n binary matrix without an all-ones 3 X 3 submatrix.

Original entry on oeis.org

1, 4, 8, 13, 20, 26, 33, 42, 49, 60, 69, 80, 92, 105, 120, 128

Offset: 1

Andrew Howroyd, Dec 24 2021

Equivalently, the maximum number of edges in a bipartite graph that is a subgraph of K_{n,n} and has no K_{3,3} induced subgraph.

			Examples of a(3)=8, a(4)=13, a(5)=20, a(6)=26:
  x x x    x x x x    x x x x .    x x x x x .
  x x x    x x x .    x x x . x    x x x x . x
  x x .    x x . x    x x . x x    x x . . x x
           x . x x    x . x x x    x . x . x x
                      . x x x x    . x . x x x
                                   . . x x x x

W. Sierpiński, Sur un problème concernant un réseau à 36 points, Ann. Soc. Polon. Math., 24: 173-174 (1951).

Wikipedia, Zarankiewicz problem

Cf. A001198, A072567, A339635, A347473, A350237.

a(n) = A001198(n) - 1 = n^2 - A350237(n) = n^2 - A347473(n) - 1.

a(14)-a(16) computed from A350237 by Max Alekseyev, Apr 01 2022, Oct 31 2022

A006616 Zarankiewicz's problem k_4(n).

Original entry on oeis.org

16, 23, 32, 43, 52, 62, 75, 87, 101, 118

Offset: 4

N. J. A. Sloane

a(n) is the least k such that every n X n {0,1}-matrix with k ones contains an all ones 4 X 4 submatrix. - Sean A. Irvine, May 17 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
R. K. Guy, A many-facetted problem of Zarankiewicz, Lect. Notes Math. 110 (1969), 129-148.
Dmitry I. Ignatov, When contranominal scales give a solution to the Zarankiewicz problem?, Workshop Notes, 12th Int'l Wksp. Formal Concept Analysis Artif. Intel. (FCA4AI 2024), 27-38. See p. 34.

Cf. A001197, A001198, A006626, A339635, A347474.

a(n) = n^2 - A347474(n). - Andrew Howroyd, Dec 26 2021

a(9)-a(13) from Andrew Howroyd, Dec 26 2021

A006620 A variant of Zarankiewicz's problem: a(n) is the least k such that every n X (n+1) {0,1}-matrix with k ones contains an all-ones 2 X 2 submatrix.

Original entry on oeis.org

5, 8, 11, 15, 19, 23, 27, 32, 37, 43, 49, 54, 59, 64

Offset: 2

N. J. A. Sloane

a(n) <= A205805(2*n+1) + 1. - Max Alekseyev, Feb 02 2024

R. K. Guy, A many-facetted problem of Zarankiewicz, Lect. Notes Math. 110 (1969), 129-148.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A006613, A006614, A006615, A006616, A006617, A006618, A006619, A006621, A006622, A006623, A006624, A006625, A006626.

Cf. A001197, A072567, A191965, A191873, A001198, A205805, A347473, A347474, A350304, A350237.

Name changed at the suggestion of Sean A. Irvine by Max Alekseyev, Feb 02 2024

A006621 Zarankiewicz's problem k_3(n,n+1).

Original entry on oeis.org

11, 17, 23, 30, 38, 46, 55, 65, 75, 87

Offset: 3

N. J. A. Sloane

a(n) is the least k such that every n X (n+1) {0,1}-matrix with k ones contains an all ones 3 X 3 submatrix. - Sean A. Irvine, May 18 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, A many-facetted problem of Zarankiewicz, Lect. Notes Math. 110 (1969), 129-148.

Cf. A001198 (k_3(n)), A006620 (k_2(n,n+1)), A006626 (k_4(n,n+1)).

a(10)-a(12) from Andrew Howroyd, Dec 26 2021

A347472 Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 2 X 2 zero submatrix.

Original entry on oeis.org

0, 2, 6, 12, 19, 27, 39, 51, 65, 81, 98, 116, 139, 163, 188, 214, 242, 272, 303, 335, 375, 413, 453

Offset: 2

M. F. Hasler, Sep 28 2021

Related to Zarankiewicz's problem k_2(n) (cf. A001197 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 2 X 2 submatrix. This complementarity leads to the given formula which was used to compute the given values.

See A347473 and A347474 for the similar problem with a 3 X 3 resp. 4 X 4 zero submatrix.

			For n = 2, there must not be any nonzero entry in an n X n = 2 X 2 matrix, if one wants a 2 X 2 zero submatrix, whence a(2) = 0.
For n = 3, having at most 2 nonzero entries in the n X n matrix still guarantees that there is a 2 X 2 zero submatrix (delete the row of the first nonzero entry and then the column of the remaining nonzero entry, if any), but if one allows 3 nonzero entries and they are placed on the diagonal, then there is no 2 X 2 zero submatrix. Hence, a(3) = 2.

Cf. A001197 (k_2(n)), A001198 (k_3(n)), A006613 - A006626.
Cf. A347473, A347474 (analog for 3 X 3 resp. 4 X 4 zero submatrix).
Cf. A350296.

a(n) = n^2 - A001197(n).

a(n) = A350296(n) - 1. - Andrew Howroyd, Dec 23 2021

a(22)-a(24) computed from A001197 by Max Alekseyev, Feb 08 2022

A006614 A variant of Zarankiewicz's problem: a(n) is the least k such that every n X n {0,1}-matrix with k ones contains an all-ones 2 X 4 submatrix.

Original entry on oeis.org

14, 21, 26, 32, 41, 48, 56, 67

Offset: 4

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
R. K. Guy, A many-facetted problem of Zarankiewicz, Lect. Notes Math. 110 (1969), 129-148.
Dmitry I. Ignatov, When contranominal scales give a solution to the Zarankiewicz problem?, Workshop Notes, 12th Int'l Wksp. Formal Concept Analysis Artif. Intel. (FCA4AI 2024), 27-38. See p. 35.

Cf. A006613, A006615, A006616, A006617, A006618, A006619, A006620, A006621, A006622, A006623, A006624, A006625, A006626.

Cf. A001197, A072567, A191965, A191873, A001198, A205805, A347473, A347474, A350304, A350237.

Name updated at the suggestion of Sean A. Irvine by Max Alekseyev, Feb 02 2024

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A001197 Zarankiewicz's problem k_2(n).

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A350237 Minimum number of 1's in an n X n binary matrix with no zero 3 X 3 submatrix.

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A347473 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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A347474 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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A350304 Maximum number of 1's in an n X n binary matrix without an all-ones 3 X 3 submatrix.

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A006616 Zarankiewicz's problem k_4(n).

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A006620 A variant of Zarankiewicz's problem: a(n) is the least k such that every n X (n+1) {0,1}-matrix with k ones contains an all-ones 2 X 2 submatrix.

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A006621 Zarankiewicz's problem k_3(n,n+1).

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A347472 Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 2 X 2 zero submatrix.

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