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Source. Ford–Fulkerson (1958), Sections 2–3, printed pp. 79–83 (published scan).

Let $A=\{a_1,\ldots,a_m\}$ be a finite set and let $\mathcal S=(S_j)_{j\in[n]}$ be an indexed family of its subsets. Different indices may carry equal sets. A representative assignment is a map $r:[n]\to A$ with $r(j)\in S_j$. Its multiplicity at $a_i$ is $c_i=|r^{-1}(\{a_i\})|$. The source calls the list of assigned elements a system of representatives.

Fix integers $0\le\alpha_i\le\beta_i$. A system of restricted representatives, or SRR, has $\alpha_i\le c_i\le\beta_i$ for every $i$. An SDR is the case $\alpha_i=0$, $\beta_i=1$. Empty indexed families and empty ground sets are included by the same formulas, with empty sums zero. In particular an empty family has an SRR precisely when every lower bound is zero.

For $X\subseteq[n]$, put

$$I_{\mathcal S}(X)=\{i:a_i\in\bigcup_{j\in X}S_j\}.$$

Write $\alpha(B)=\sum_{i\in B}\alpha_i$ and $\beta(B)=\sum_{i\in B}\beta_i$ for $B\subseteq[m]$, and set $a=\alpha([m])$. The letter $a$ without a subscript is this sum, not an element of the ground set.

A second family $\mathcal T=(T_j)_{j\in[n]}$ has the same number of indices. A common SRR consists of an assignment for each family with the same multiplicities $c_i$, satisfying the same bounds. The assignments may put the same element at different indices: a common multiset of representatives need not be a coordinatewise representative of every intersection $S_j\cap T_j$.

The integrality of the bounds is needed for the source's use of integral flows. If real bounds are specified, first replace them by $\lceil\alpha_i\rceil$ and $\lfloor\beta_i\rfloor$ and check their order. Substituting arbitrary real bounds directly into the printed tests is insufficient. For example, one set $\{a_1,a_2\}$ with zero lower bounds and both upper bounds $1/2$ satisfies the unrounded Theorem 1 inequalities but has no representative obeying those bounds. This is an explicit interpretation of occurrence bounds, not a claim that the source announced a theorem for nonintegral restrictions.

All network vertices below are tagged by layer. Thus set indices, element labels, and the source and sink are distinct vertices even when their written labels happen to agree.