1,3

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.

a(9) >= 45. - Richard Bean (rwb(AT)eskimo.com), May 01 2002

For n sufficiently large (>= 295), a(n) >= (n^2)*(1-(2 + ln 2)/ln n) + n*(1 + (ln(8*pi)/ln n) - (ln 2}/(ln n). Bean and Mahmoodian also show a(n) <= n^2 - 3n + 3. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 03 2007

Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, Discrete Math., 267 (2003), 13-21.

R. Bean and Ebadollah S. Mahmoodian, A new bound on the size of the largest critical set in Latin squares, Discrete Math, to appear.

Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square

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, Bulletin of the Institute of Combinatorics and its Applications (Canada). 38 (2003) pp. 19-22

Sequence in context: A072456 A138659 A020590 this_sequence A049792 A062851 A071979

Adjacent sequences: A063434 A063435 A063436 this_sequence A063438 A063439 A063440

nonn

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 24 2001

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.