On a Combinatorial Problem in Latin Squares

1. Denote by S, an arbitrary latin square with n elements {a 1 , a 2 , . . . , a„ } . A row and column of this square, intersecting on the main diagonal (i .e . diagonal beginning at the left lower corner) will be called corresponding . After striking out n c arbitrary rows and corresponding columns, a square T, with c x c entries remains . Such a square will be called a principal minor. It is clearly determined by denoting its c elements belonging to the main diagonal . Denote by ki, i . . . iQ (i i , i 2, . . . , ig = 1, 2, . . . , n all different) the number of columns in T, containing the elements ai,, aip , . . . , ai q simultaneously . Let 0) be the minimum of ki„ i„ ..., iq . We shall consider the following problem : Assuming that n and k (q) are two given positive integers, what is the minimal c (denoted by b), such that from an arbitrary S, at least one T, can be obtained with the prescribed k(q . The problem is solved by a method used already in [1] and [2] . The question for the case of k(2 ) arises in connection with so called generalized normal multiplication tables of groups (and other systems) [3], [4], 15]. Such tables are complete (i .e . the product of any two group elements appears explicitly in them) if and only if k(2 > 1 . E.g. the following