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For µ given latin squares of order n, they have k intersection when they have k identical cells and n 2 − k cells with mutually different entries. For each n ≥ 1 the set of integers k such that there exist µ latin squares of order n with k intersection is denoted by I µ [n]. In a paper by P. , I 3 [n] is determined completely. In this paper we completely determine I 4 [n] for n ≥ 16. For n ≤ 16, we find out most of the elements of I 4 [n].

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A µ-way latin trade of volume s is a set of µ partial latin rectangles (of inconsequential size) containing exactly the same s filled cells, such that if cell (i, j) is filled, it contains a different entry in each of the µ partial latin rectangles, and such that row i in each of the µ partial latin rectangles contains, set-wise, the same symbols and column j, likewise. In this paper we show that all µ-way latin trades with sufficiently large volumes exist, and state some theorems on the non-existence of µ-way latin trades of certain volumes. We also find the set of possible volumes (that is, the volume spectrum) of µ-way latin trades for µ = 4 and 5. (The case µ = 2 was dealt with by Fu, and the case µ = 3 by the present authors.) (2000). 05B15. Mathematics Subject Classification