A001197 -id:A001197 - cp's OEIS frontend

A006625 A variant of Zarankiewicz's problem: a(n) is the least k such that every n X (n+2) {0,1}-matrix with k ones contains an all-ones 3 X 4 submatrix.

14, 21, 28, 36, 45, 55

Offset: 3

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, A006620, A006621, A006622, A006623, A006624, A006626.

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

Name changed at the suggestion of Sean A. Irvine and a(8) added by Max Alekseyev, Feb 02 2024

A001841 Related to Zarankiewicz's problem.

Original entry on oeis.org

3, 5, 10, 14, 21, 26, 36, 43, 55, 64, 78, 88, 105, 117, 136, 150, 171, 186, 210, 227, 253, 272, 300, 320, 351, 373, 406, 430, 465, 490, 528, 555, 595, 624, 666, 696, 741

Offset: 3

N. J. A. Sloane

nonn

Definition appears to be: a(n) is the maximum number of triangles in K_n, where each edge may be used 3 times. - Charles R Greathouse IV, Jul 06 2017

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, (p. 126, divided by 2).
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).

John Cerkan, Table of n, a(n) for n = 3..10000
R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. See p. 9 column t(3,m). [Annotated and scanned copy, with permission]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

A001841:=-(2*z**4+z**5+2*z**2+2*z**3+2*z+3)/(z**2-z+1)/(z**2+z+1)/(z+1)**2/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

A004037 The coding-theoretic function A(n,6,4).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 9, 13, 14, 15, 20, 20, 22, 25, 30, 31, 37, 40, 42, 50, 52, 54, 63, 65, 67, 76, 80, 82, 92, 96, 99, 111, 114, 117, 130, 133, 136, 149, 154, 157, 171, 176, 180, 196, 200, 204, 221, 225, 229, 246, 252, 256, 274, 280, 285, 305, 310, 315, 336

Offset: 4

N. J. A. Sloane

nonn

Packing number D(n,4,2). Maximum number of edge-disjoint K_4's in a K_n. - Rob Pratt, Feb 26 2024

CRC Handbook of Combinatorial Designs, 1996, p. 411.
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).

A. E. Brouwer, Bounds for binary constant weight codes
A. E. Brouwer, Optimal packings of K_4's into a K_n, J. Combin. Theory, A 26 (1979), 278-297.
A. E. Brouwer, A. Schrijver, and H. Hanani, Group divisible designs with block-size four, Discrete Math. 20 (1977/78), no. 1, 1-10.
A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Department of Mathematics, University of Calgary, January 1967. [Annotated and scanned copy, with permission]
Sascha Kurz, Plane point sets with many squares or isosceles right triangles, arXiv:2112.12716 [math.CO], 2021.
Alice Miller and Michael Codish, Graphs with girth at least 5 with orders between 20 and 32, arXiv:1708.06576 [math.CO], 2017, Table 3.
Index entries for sequences related to A(n,d,w)

Offset corrected by Zhao Hui Du, Aug 14 2008
Offset 4 and a(4) inserted by Jeremy Tan, Feb 16 2022
a(56)-a(64) from Rob Pratt, Feb 27 2024

A376167 Square array read by antidiagonals: T(n,k) = smallest r such that every n X k binary matrix with r ones contains a 2 X 2 submatrix of ones, with n, k >= 2.

Original entry on oeis.org

4, 5, 5, 6, 7, 6, 7, 8, 8, 7, 8, 9, 10, 9, 8, 9, 10, 11, 11, 10, 9, 10, 11, 13, 13, 13, 11, 10, 11, 12, 14, 15, 15, 14, 12, 11, 12, 13, 15, 16, 17, 16, 15, 13, 12, 13, 14, 16, 18, 19, 19, 18, 16, 14, 13, 14, 15, 17, 19, 20, 22, 20, 19, 17, 15, 14, 15, 16, 18, 21, 22, 23, 23, 22, 21, 18, 16, 15

Offset: 2

Paolo Xausa, Sep 13 2024

			Array begins (cf. Table 5 in Knuth (2023), section 7.2.2.2, p. 291, where T(n,k) is denoted by Z(m,n)):
  n\k|   2   3   4   5   6   7   8   9  10  11  12  ...
  -----------------------------------------------------
   2 |   4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, ...
   3 |   5,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, ...
   4 |   6,  8, 10, 11, 13, 14, 15, 16, 17, 18, 19, ...
   5 |   7,  9, 11, 13, 15, 16, 18, 19, 21, 22, 23, ...
   6 |   8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, ...
   7 |   9, 11, 14, 16, 19, 22, 23, 25, 26, 28, 29, ...
   8 |  10, 12, 15, 18, 20, 23, 25, 27, 29, 31, 33, ...
   9 |  11, 13, 16, 19, 22, 25, 27, 30, 32, 34, 37, ...
  10 |  12, 14, 17, 21, 23, 26, 29, 32, 35, 37, 40, ...
  11 |  13, 15, 18, 22, 25, 28, 31, 34, 37, 40, 43, ...
  12 |  14, 16, 19, 23, 26, 29, 33, 37, 40, 43, 46, ...
  ...
T(3,4) = 8 because, no matter how 8 ones are arranged in a 3 X 4 matrix, a 2 X 2 submatrix of ones cannot be avoided (in the left configuration below, for example, the submatrix elements are highlighted by parentheses). 7 ones can avoid such submatrix (right).
.
  (1) 0 (1) 1       1  0  1  1
   0  1  0  1       0  1  0  1
  (1) 1 (1) 0       1  1  0  0

Donald E. Knuth, The Art of Computer Programming, Vol. 4B: Combinatorial Algorithms, Part 2, Addison-Wesley, 2023, section 7.2.2.2, pp. 289-291.

Cf. A001197 (main diagonal), A347472, A350296.

T(n,k) = T(k,n).
T(2,k) = k + 2.
T(n,2) = n + 2.

A351100 Maximum number of 4-subsets of an n-set such that every 3-subset is covered at most twice.

Original entry on oeis.org

2, 5, 9, 15, 28, 40, 60, 80, 108, 143, 182, 225, 280, 340, 405

Offset: 4

Jeremy Tan, Jan 31 2022

Maximum number of K_4^3's that can be packed in a doubled K_n^3, where K_n^m is the complete m-uniform hypergraph on n vertices.

			a(6) = 9 because of the following optimal collection of 4-subsets:
  1 2 3 4
  2 3 4 5
  3 4 5 6
  4 5 6 1
  5 6 1 2
  6 1 2 3
  1 2 4 5
  2 3 5 6
  3 4 6 1

Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Department of Mathematics, University of Calgary, January 1967. [Annotated and scanned copy, with permission]
Haim Hanani, On quadruple systems, Canadian Journal of Mathematics, 12 (1960), 145-157.
Jeremy Tan, An attack on Zarankiewicz's problem through SAT solving, arXiv:2203.02283 [math.CO], 2022.

Cf. A001839-A001843 for other packing sequences discussed in Richard K. Guy's paper.

a(n) >= 2*A001843(n). Equality holds if n = 6k+2 or 6k+4 (Hanani).

cp's OEIS Frontend

A006625 A variant of Zarankiewicz's problem: a(n) is the least k such that every n X (n+2) {0,1}-matrix with k ones contains an all-ones 3 X 4 submatrix.

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A001841 Related to Zarankiewicz's problem.

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A004037 The coding-theoretic function A(n,6,4).

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A376167 Square array read by antidiagonals: T(n,k) = smallest r such that every n X k binary matrix with r ones contains a 2 X 2 submatrix of ones, with n, k >= 2.

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A351100 Maximum number of 4-subsets of an n-set such that every 3-subset is covered at most twice.

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