|
|
A001198
|
|
Zarankiewicz's problem k_3(n).
(Formerly M4601 N1962)
|
|
13
|
|
|
9, 14, 21, 27, 34, 43, 50, 61, 70, 81, 93, 106, 121, 129
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
3,1
|
|
COMMENTS
|
Guy denotes k_{a,b}(m,n) the least k such that any m X n matrix with k '1's and '0's elsewhere has an a X b submatrix with all '1's, and omits b (resp. n) when b = a (resp. n = m). With this notation, a(n) = k_3(n). Sierpiński (1951) found a(4..6), a(7) is due to Brzeziński and a(8) due to Čulík (1956). - M. F. Hasler, Sep 28 2021
|
|
REFERENCES
|
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.
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).
|
|
LINKS
|
R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
|
|
FORMULA
|
|
|
EXAMPLE
|
For n = 3 it is clearly necessary and sufficient that there be 3 X 3 = 9 ones in the n X n matrix in order to have an all-ones 3 X 3 submatrix.
For n = 4 there may be at most 2 zeros in the 4 X 4 matrix in order to be guaranteed to have a 3 X 3 submatrix with all '1's, whence a(4) = 16 - 2 = 14: If 3 zeros are placed on a diagonal, it is no more possible to find a 3 X 3 all-ones submatrix, but if there are at most 2 zeros, one always has such a submatrix, as one can see from the following two diagrams:
0 1 1 1 0 1 1 1 no 3 X 3
Here one can delete, e.g., -> 1 0 1 1 1 0 1 1 <- all-ones
row 1 and column 2 to get 1 1 1 1 1 1 0 1 submatrix
an all-ones 3 X 3 matrix. 1 1 1 1 1 1 1 1 (End)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,more,hard
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|