OFFSET
1,3
COMMENTS
A critical set in an n X n array is a set C of given entries such that there exists a unique extension of C to an n X n Latin square and no proper subset of C has this property.
The next terms satisfy a(7) >= 25, a(8) >= 37, a(9) >= 44, a(10) >= 57. In the reference it is proved that, for all n, a(n) <= n^2 - 3n + 3.
a(9) >= 45. - Richard Bean, May 01 2002
a(9) >= 46, a(10) >= 59, a(11) >= 74, a(12) >= 90, a(13) >= 105, a(14) >= 126, a(15) >= 147, a(16) >= 175, a(17) >= 198. - Richard Bean, May 13 2026
For n sufficiently large (>= 295), a(n) >= (n^2)*(1-(2 + log 2)/log n) + n*(1 + log(8*Pi)/log n) - (log 2)/(log n). Bean and Mahmoodian also show a(n) <= n^2 - 3n + 3. - Jonathan Vos Post, Jan 03 2007
LINKS
Richard Bean, New examples for a(9)-a(17) bounds
Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, arXiv:math/0107159 [math.CO], 2001.
Richard Bean and Ebadollah S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, Discrete Math., 267 (2003), 13-21.
Mahya Ghandehari, Hamed Hatami and Ebadollah S. Mahmoodian, On the size of the minimum critical set of a Latin square, arXiv:math/0701015 [math.CO], 2006.
Mahya Ghandehari, Hamed Hatami and Ebadollah S. Mahmoodian, On the size of the minimum critical set of a Latin square, Journal of Discrete Mathematics. 293(1-3) (2005) pp. 121-127.
Hamed Hatami and Ebadollah S. Mahmoodian, A lower bound for the size of the largest critical sets in Latin squares, arXiv:math/0701014 [math.CO], 2006; Bulletin of the Institute of Combinatorics and its Applications (Canada). 38 (2003) pp. 19-22
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 24 2001
EXTENSIONS
a(7) from Richard Bean, Mar 06 2026
STATUS
approved