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Statement

Conventions (pp. 111--112, as on the Theorem 1 page); type $\omega$ means the set-mapping is defined on the finite subsets. Theorems marked (**) use the hypothesis (**) of p. 112: for a strongly inaccessible cardinal $m$ and a set $S$ of power $m$ there is a two-valued measure $\mu$ on all subsets of $S$ with $\mu(S)=1$, $\mu(\{x\})=0$ for every $x\in S$, and additive for fewer than $m$ summands. The paper does not examine whether its theorems are equivalent to this hypothesis.

Theorem 7 (p. 123, quoted). "(**) If the cardinal number $\aleph_\alpha>\aleph_0$ is strongly inaccessible, then $(\aleph_\alpha, \aleph_\beta, \omega)\to\aleph_\alpha$ for every $\beta<\alpha$."

So, under (**) for $\aleph_\alpha$, every set-mapping of a set of power $\aleph_\alpha$ defined on its finite subsets, with all values of power less than some $\aleph_\beta<\aleph_\alpha$, has a free set of power $\aleph_\alpha$. Section 3 (p. 113) calls this result surprising. Theorem 8 (p. 125), a consequence of Theorems 6 and 7, gives under (**) $(\aleph_\alpha,\aleph_\beta,k)\to\aleph_\alpha$ for every limit ordinal $\alpha$, every $\beta<\alpha$ and $k=1,2,\ldots$.

Source. P. Erdős and A. Hajnal, On the structure of set-mappings, Acta Math. Acad. Sci. Hungar. 9 (1958), 111--131: Theorem 7 on p. 123, proof pp. 123--125, announced on p. 113; hypothesis (**) on p. 112; Theorem 8 on p. 125. The edition is the one identified on the source card.

Read depth. Claims checked: the statements of Theorems 7 and 8 and the hypothesis (**) were read clause by clause on the printed pages. The proof was not checked.

Proof pointer

The proof (pp. 123--125) passes from $f$ to its derived set-mapping, which puts $y$ into the value of a finite set $A$ when the set of $x$ with $y\in f(A\cup\{x\})$ has measure 1, and shows that the derived mapping and its iterates keep order $\aleph_\beta$. A sequence of length $\omega_\alpha$ is then chosen avoiding the measure-zero sets these mappings define, and a point set-mapping built from them, of order $\aleph_{\beta+1}$ (and $\aleph_{\beta+1}<\aleph_\alpha$), has a free set of power $\aleph_\alpha$ by a theorem of Erdős that the paper cites from its reference [1] (P. Erdős, Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950), 127--141); that set is shown to be free for $f$.

Dependencies

Hypothesis (**) for $\aleph_\alpha$; the free-set theorem for set-mappings of type 1 that the paper cites from Erdős 1950.

Bears on

No Erdős problem page directly. It is a positive result for type $\omega$ at strongly inaccessible cardinals carrying such a measure, far above the cardinal $\aleph_\omega$ of the paper's Problem 1.