Sets of lattice points which contain a maximal number of edges

Abstract

How should one select an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -element subset of a rectangular array of lattice points (points with integral coordinates) in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Euclidean space so as to include the largest possible number of edges (pairs of points differing in exactly one coordinate)? It is shown that the generalized Macaulay theorem due to the author and B. Lindström contains the (known) solution.