A new bound on the size of the largest 2-critical set in a latin square

A critical set is a partial latin square that has a unique completion to a latin square of the same order, and is minimal in this property. If P is a critical set in a latin square L, then each element of P must be contained in a latin trade Q in L such that |P ∩ Q| = 1. In the case where each element of P is contained in an intercalate (latin trade of size 4) Q such that |P∩Q| = 1 we say that P is 2-critical. In this paper we show that the size of a 2-critical set in a latin square L is no greater than n −O(n5/4). 1 Background information In any combinatorial configuration it is possible to identify a subset which uniquely determines the structure of the configuration and in some cases is minimal with respect to this property. Examples of such subsets can be found by studying the literature on critical sets in latin squares (see Donovan and Howse [8]) and defining sets in block designs (see Street [11]). The recent research in these areas has focused on building a bank of knowledge which may be used to determine the spectrum of the prescribed subsets. With this current paper we restrict ourselves to a discussion of critical sets in latin squares. We define scs(n) and lcs(n) to be the sizes of the smallest and largest critical sets in any latin square of order n. The problem of determining these values exactly for every n remains unsolved. However progress has been made in both cases. Fu, Fu and Rodger ([9]) showed that if n > 20, scs(n)≥ (7n−3)/6 . The smallest critical set so far constructed for any latin square of size n has size n/4 ([5], [4]). A critical set of such size is known to exist in back circulant latin squares, namely those latin squares based on the addition table for the integers modulo n. Australasian Journal of Combinatorics 26(2002), pp.255–263 Donovan, Cooper, Nott and Seberry ([7]) examined lower bounds for critical sets of latin squares based on certain groups. Bate and van Rees ([1]) showed that the size of the smallest strong critical set (a critical set with a certain type of completion) is n/4 . Stinson and van Rees ([10]) determined some lower bounds for lcs(n) for small values of n. The largest critical sets that have been constructed in latin squares of order n have size 4 − 3, where n = 2. It is interesting to note that these critical sets are also 2-critical. This has lead to a conjecture made by Bean and Mahmoodian [3]. Conjecture 1 lcs(n) ≤ n − n2 3 ≈ n − n. A larger upper bound for lcs(n) is conjectured by Brankovic, Horak, Miller and Rosa [2]. Conjecture 2 lcs(n) ≤ n − n. Curran and van Rees [5] showed that lcs(n) ≤ n − n; this bound was improved by Bean and Mahmoodian [3]: Theorem 3 lcs(n) ≤ n − 3n+ 3. This is currently the best known upper bound that applies to arbitrary critical sets. In Theorem 11 we find an improved upper bound for 2-critical sets.