Latin squares from multiplication tables

Abstract

AbstractConsider the usual multiplication table in which all entries strictly greater than were deleted. Can we fill in the empty places so that the resulting table will become a multiplication table of a group? This intriguing question arose in connection to the famous Graham's Greatest Common Divisor Problem. We prove here a weaker statement, namely, that one may always complete such a partial multiplication table to a Latin square. Our result is more general, as it guarantees the possibility of completion of any partial Latin square, provided that the shape of the empty part satisfies certain conditions. We also point on connections of our result to a far‐reaching extension of Graham's problem involving graphs and homogeneous arithmetic progressions.