Every SOMA(n-2, n) is Trojan

Trivially, a SOMA(k, n) is Trojan for k = 0 and k = 1. Moreover, R. A. Bailey proved that every SOMA(n − 1, n) is Trojan. Computational analysis of some examples suggested that a similar result held for k = n − 2. This led Bailey, Cameron and Soicher to ask (BCC Problem 16.19 in Discrete Math. Vol. 197/198) whether a SOMA(n − 2, n) must be Trojan. 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 and 7. While the result of Bailey shows that a 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 the existence of a SOMA(n − 2, n) is equivalent to the existence of an affine plane of order n, when n ≥ 5.