An inequality for pseudo-subplanes of sets of orthogonal latin squares

AbstractLet L1, L2,…, Lt be a given set of t mutually orthogonal order-n latin squares defined on a symbol set S, |S| = n. The squares are equivalent to a (t + 2)-netN of order n which has n2 points corresponding to the n2 cells of the squares. A line of the net N defined by the latin square Li comprises the n points of the net which are specified by a set of n cells of Li all of which contain the same symbol x of S. If we pick out a particular r × r block B of cells, a line which contains points corresponding to r of the cells of B will be called an r-cell line. If there exist r(r − 1) such lines among the tn lines of N, we shall say that they form a pseudo-subplane of order r-the “pseudo” means that these lines need not belong to only r − 1 of the latin squares. The purpose of the present note is to prove that the hypothesis that such a pseudo-plane exists in N implies that r3 − (t + 2)r2 + r + nt ⩾∗0