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%I #18 Mar 29 2019 07:30:13
%S 0,1,3,7,11,18
%N Cardinality of largest critical set in any Latin square of order n.
%C 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.
%C 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.
%C a(9) >= 45. - _Richard Bean_, May 01 2002
%C 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
%H Richard Bean and E. S. Mahmoodian, <a href="https://arxiv.org/abs/math/0107159">A new bound on the size of the largest critical set in a Latin square</a>, arXiv:math/0107159 [math.CO], 2001.
%H Richard Bean and Ebadollah S. Mahmoodian, <a href="https://doi.org/10.1016/S0012-365X(02)00599-X">A new bound on the size of the largest critical set in a Latin square</a>, Discrete Math., 267 (2003), 13-21.
%H Mahya Ghandehari, Hamed Hatami and Ebadollah S. Mahmoodian, <a href="https://arxiv.org/abs/math/0701015">On the size of the minimum critical set of a Latin square</a>, arXiv:math/0701015 [math.CO], 2006.
%H Mahya Ghandehari, Hamed Hatami and Ebadollah S. Mahmoodian, <a href="https://doi.org/10.1016/S0012-365X(02)00599-X">On the size of the minimum critical set of a Latin square</a>, Journal of Discrete Mathematics. 293(1-3) (2005) pp. 121-127.
%H Hamed Hatami and Ebadollah S. Mahmoodian, <a href="https://arxiv.org/abs/math/0701014">A lower bound for the size of the largest critical sets in Latin squares</a>, arXiv:math/0701014 [math.CO], 2006; Bulletin of the Institute of Combinatorics and its Applications (Canada). 38 (2003) pp. 19-22
%K nonn,more
%O 1,3
%A Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 24 2001
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