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Source. Erdős (1945), printed pp. 898 and 900–901 (published scan).

Theorem 1 invokes Sperner's theorem: an antichain of distinct subsets of $[N]$ has size at most $\binom N{\lfloor N/2\rfloor}$. The source cites E. Sperner, Mathematische Zeitschrift 27 (1928), 544–548. Its original proof is not reproduced from that separate paper. Within this source unit, Theorem 4 at $r=1$ proves exactly the needed antichain bound by the source's shadow method.

For Theorem 5, Erdős quotes Menger's vertex-disjoint-path theorem through König's graph-theory book. The displayed counting argument and subsequent compression need paths that increase rank at every step. The path lemma therefore orients Boolean-lattice cover edges upward and makes the directed interface explicit.

The exact later input used for this expansion is Ford–Fulkerson's finite integral-flow theorem: in a finite directed network with nonnegative integer capacities, distinct terminals $s,t$, no arcs into $s$ and no arcs out of $t$, a maximum flow exists, its value is the minimum outgoing cut capacity, and an integral maximum exists. An outgoing cut is the set of arcs from $X$ to its complement, where $s\in X$ and $t\notin X$.

Ford–Fulkerson's node-splitting construction also gives the vertex-capacity interpretation. The path lemma writes out its own finite split network, its cut correspondence and the unit-path decomposition of an integral flow. All capacities are finite integers, and the network is acyclic. No termination statement for irrational capacities, infinite graph theorem or unproved path-orientation assumption is used.

The complete Ford–Fulkerson proof lives at the linked source; it is an external input here. This later implementation of the directed Menger interface is a compilation expansion, not an attribution of a 1957 result to the 1945 paper.

The introduction's Littlewood–Offord predecessor estimate is historical context, not an input to the proofs compiled here. The later symmetric-chain antichain bound is a distinct proof and is not substituted for Erdős's shadow or Menger arguments.