A critical set in an n × n Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is ⌊n 2 /4⌋. We prove a lower-bound of n 2 /10 4 for sufficiently large n, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lower-bound of Ω(n 3/2 ) due to Cavenagh and Ramadurai.From the point of view of computational learning theory, the size of the smallest critical set corresponds to the minimum teaching dimension of the set of Latin squares. We study two related notions of dimension from learning theory. We prove a lower-bound of n 2 −(e+o(1))n 5/3for both of the VC-dimension and the recursive teaching dimension.

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For a graph H, letwhere the maximum is taken over all graphs G on n vertices not containing H as a minor. Thus c ∞ (H) is the asymptotic maximum density of graphs not containing H as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on c ∞ (H) for disconnected graphs H. In particular, we determine c ∞ (H) whenever H is union of cycles. Finally, we investigate the behaviour of c ∞ (sK r ) for fixed r, where sK r denotes the union of s disjoint copies of the complete graph on r vertices. Improving on a result of Thomason, we show that c ∞ (sK r ) = s(r − 1) − 1 for s = Ω log r log log r , and c ∞ (sK r ) > s(r − 1) − 1 for s = o log r log log r .

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