The normalized matching property from the generalized Macaulay theorem (Q1904384)

The normalized matching property for multisets asserts that the density of a family \(\mathcal A\) of \(k\)-element subsets of a multiset \(M\) in the family of all \(k\)-element subsets never exceeds the density of its shadow \(\Delta{\mathcal A}\) in the \((k-1)\)-element subsets. It is shown that this well-established result can be obtained as a corollary to the generalized Macaulay theorem of the author and \textit{B. Lindström} [J. Comb. Theory 7, 230-238 (1969; Zbl 0186.01701)].