Every is Trojan

A SOMA ( k , n ) is an n × n array A each of whose entries is a k -subset of a k n -set Ω of symbols, such that every symbol of Ω occurs exactly once in each row and exactly once in each column of A , and every 2-subset of Ω is contained in at most one entry of A . A SOMA ( k , n ) is called Trojan if it can be constructed by the superposition of k mutually orthogonal Latin squares of order n with pairwise disjoint symbol-sets. Note that not every SOMA ( k , n ) is Trojan, and if k ≥ n then there exists no SOMA ( k , n ) . Trivially, every SOMA ( 0 , n ) and every SOMA ( 1 , n ) is Trojan. R. A. Bailey proved that every SOMA ( n − 1 , n ) is Trojan. Bailey, Cameron and Soicher then asked whether a SOMA ( n − 2 , n ) must be Trojan, which is posed in B. C. C. Problem 16.19 in Discrete Math. vol. 197/198. In this paper, we prove that this is indeed the case. We remark that there are non-Trojan SOMA ( n − 3 , n ) s, at least when n = 5 , 6 , 7 . While the result of Bailey shows that the existence of a SOMA ( n − 1 , n ) is equivalent to the existence of an affine plane of order n , our result together with known results show that if n ≥ 5 then the existence of a SOMA ( n − 2 , n ) is equivalent to the existence of an affine plane of order n .