In order for the family \(\mathcal{A} = (A_1, A_2 ,\ldots , A_n)\) of sets to have an \(SDR\), it is necessary that the following condition hold:
(MC): For each \(k=1,2, \ldots , n\) and each choice of \(k\) distinct indices \(i_1, i_2, \ldots , i_k\) from \(\{ 1,2, \ldots , n\} \),

in short, every \(k\) sets of the family collectively contain at least \(k\) elements.

The family \(\mathcal{A} = (A_1, A_2 ,\ldots , A_n)\) of subsets of a set \(Y\) has an SDR if and only if the marriage condition MC holds.

Let \(\mathcal{A} = (A_1, A_2 ,\ldots , A_n)\) be a family of subsets of a finite set \(Y\). Let \(t\) be an integer with \(0 \leq t \leq n\). Then there exists a subfamily of \(t\) sets of \(\mathcal{A}\) that has an SDR if and only if

for all \(k\) with \(k \geq n-t\) and all choices of \(k\) distinct indices \(i_1, i_2, \ldots , i_k\) from \(\{ 1,2, \ldots , n\} \).

Let \(\mathcal{A} = (A_1, A_2 ,\ldots , A_n)\) be a family of subsets of a finite set \(Y\). Then the largest number of sets in a subfamily of \(\mathcal{A}\) with an SDR equals the smallest value taken by the expression

over all choices of \(k = 1,2, \ldots ,n\) and all choices of \(k\) indices \(i_1, i_2, \ldots , i_k\) with \(1 \leq i_1 {\lt} i_2 {\lt}\ldots {\lt} i_k \leq n\).

For each preferential ranking matrix, there exists a stable complete marriage.

The stable complete marriage obtained from the deferred acceptance algorithm, with the women choosing the men, is women-optimal. If the men choose the women in the deferred acceptance algorithm, the resulting complete marriage is men-optimal.