Consider the usual multiplication table $n\times n$ in which all entries strictly greater than $n$ 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.

中文翻译：

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乘法表中的拉丁方

考虑通常的乘法表 $ñ\times ñ$ 其中所有条目严格大于 $ñ$被删除。我们可以填写空白处，以使结果表成为组的乘法表吗？这个有趣的问题是与著名的格雷厄姆的“最大除数问题”有关的。我们在这里证明了一种较弱的说法，即，人们总是可以将这样的部分乘法表填写成一个拉丁方。我们的结果更为笼统，因为只要空白部分的形状满足某些条件，它就可以保证完成任何部分拉丁方的可能性。我们还指出了我们的结果与涉及图和齐次算术级数的Graham问题的深远扩展之间的联系。