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Statement

Conventions (p. 111). A set-mapping of type 1 on $S$ assigns to each point $x\in S$ a set $f(x)\subseteq S$ with $x\notin f(x)$ (the paper's original notion, a mapping on the one-element subsets); order $n$ means $\lvert f(x)\rvert<n$ for every $x$. A set $S'\subseteq S$ is free when $y\notin f(x)$ for all $x,y\in S'$.

Lemma 4 (p. 119). Let $S$ be a set of power $m\ge\aleph_0$, let $n<m$, and let $f$ be a set-mapping of $S$ of type 1 and order $n$. Then $S$ is the union of at most $n$ free sets.

Source. P. Erdős and A. Hajnal, On the structure of set-mappings, Acta Math. Acad. Sci. Hungar. 9 (1958), 111--131: Lemma 4 on p. 119, with footnote 9 citing G. Fodor, Proof of a conjecture of P. Erdős, Acta Sci. Math. Szeged 14 (1951--1952), 219--227, Theorem 1. The edition is the one identified on the source card.

Read depth. Claims checked: the statement and the definitions it uses were read on the printed pages. The paper gives no proof; it cites Fodor's theorem.

Proof pointer

No proof in the paper: Lemma 4 is stated as a theorem of G. Fodor, with the reference above. The paper uses it in the proof of Theorem 3 (p. 119).

Bears on

No Erdős problem page directly; the paper uses it as a tool for Theorem 3.