Kunen 2013 impact paul erdos set theory

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theorem_p348_erdos_tarski: The survey's modern restatement of the Erdős-Tarski properties P_1 to P_4, Q and R of a cardinal λ > ω, with the results it reports: each P_m implies P_{m+1} for m = 1, 2, 3, not-P_1 implies λ strongly inaccessible, and for strongly inaccessible λ the negations of P_1, P_2, Q, R are each equivalent to weak compactness, not-P_3 to measurability and not-P_4 to strong compactness.

theorem_p350_erdos_hajnal: The survey's report of Erdős and Hajnal's equivalent of weak compactness, a variant of the free set lemma: for an infinite cardinal κ, every family of κ pairwise incomparable subsets of κ of size below κ has a subfamily of size κ whose union misses at least κ points if and only if κ = ω or κ is weakly compact.

theorem_p359_free_set_lemma: The survey's statement of Hajnal's Free Set Lemma, that for infinite cardinals κ < λ every g from λ into the subsets of λ of size below κ has a free set of size λ, with its history (Lázár for regular λ, Erdős for singular λ under GCH) and the example g(α) = α showing that the bound |g(α)| < λ alone is not enough.

theorem_p359_nowhere_dense: The survey's report of Erdős's 1954 results on set mappings g of the reals with small images: no free set of size 2 is guaranteed when small means of size below c or not dense, size 2 but not 3 when it means both, a free set of size ℵ_0 when every g(x) is nowhere dense (Theorem 6), an everywhere dense one by Bagemihl, and independence of a free set of size ℵ_1.

theorem_p360_erdos_hajnal_mate: The survey's report of Theorem 3.8 of Erdős, Hajnal and Máté: for regular λ and g from λ into the subsets of λ whose range satisfies Condition B, there is a free set of size ℵ_0, one of size μ when μ < λ and ν^{<μ} < λ for all ν < λ, and one of size λ when λ is weakly compact, which a λ-Suslin tree shows cannot be weakened to strongly inaccessible.

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Kenneth Kunen, The Impact of Paul Erdős on Set Theory. Erdős Centennial, Bolyai Society Mathematical Studies 25, Springer (2013), pp. 347-363. doi:10.1007/978-3-642-39286-3_12. No notice is printed in the file, the publisher's typeset chapter (printed pp. 347--348 and 362--363 carry no copyright or license line); the publisher's chapter page (https://link.springer.com/chapter/10.1007/978-3-642-39286-3_12, read 2026-10-02) states "© 2013 János Bolyai Mathematical Society and Springer-Verlag" and offers the chapter behind a paywall with "Reprints and permissions" and no Creative Commons statement, every other right reserved.

Kunen surveys the areas of set theory where Erdos's influence is still felt, deliberately omitting partition and graph theory (covered by Komjath's chapter in the same volume). Section 2 recasts the Erdos-Tarski properties P1-P4, Q, R in modern terms, noting that for strongly inaccessible lambda the negations of P1, P2, Q, R are all equivalent to weak compactness while not-P3 is measurability and not-P4 strong compactness. Later sections cover chain conditions in forcing, set-theoretic topology, order types, and Erdos's Euclidean partition results. Section 8, 'Free Sets' (pp. 359--360), traces the Free Set Lemma from Lazar (1936, regular lambda) through Erdos (1950, singular lambda under GCH) to Hajnal's ZFC proof (1960), discusses 'Some remarks on set theory III' (1954), where, depending on the sense in which each g(x) is small, there may be no free set of size 2, a free set of size 2 but possibly none of size 3, or (Erdos's Theorem 6, g(x) nowhere dense) a free set of size aleph_0, which Bagemihl improved to an everywhere dense free set, and notes that for nowhere-dense g(x) whether a free set of size aleph_1 or more must exist is independent. It then reports Erdos-Hajnal-Mate (1973), whose Theorem 3.8 obtains free sets for regular lambda under Condition B on ran(g) (which Condition A implies), including a free set of size lambda when lambda is weakly compact; a lambda-Suslin tree gives a counterexample to that clause with weakly compact replaced by strongly inaccessible (p. 360). The chapter is survey material only: it proves nothing, and it does not mention the finite function H(n) of #624 or the 1968 paper behind it; its Section 8 covers free sets for set mappings on infinite sets.

Source: https://doi.org/10.1007/978-3-642-39286-3_12.

Read status: claims checked for the statements on the result pages below, read clause by clause on the page images of the print (pp. 348--350 and 359--360). The chapter is a survey and proves none of them; the papers it cites are not held and were not read. Nothing here is independently reviewed. Result pages: theorem_p348_erdos_tarski, theorem_p350_erdos_hajnal, theorem_p359_free_set_lemma, theorem_p359_nowhere_dense and theorem_p360_erdos_hajnal_mate.

Bears on. #624: context only; Section 8 surveys free sets for set mappings on infinite sets, and the chapter does not mention the problem's finite function $H(n)$ and decides nothing about it. #501: context only; the report on Erdős's 1954 paper (p. 359) treats set mappings of the reals with images small in cardinality or density, not bounded with outer measure below $1$ as the problem asks, and decides nothing about it. #1173: context only; the Free Set Lemma (p. 359) needs a bound $\kappa<\lambda$ on the sizes of the images, which the problem's mappings do not have, and the example $g(\alpha)=\alpha$ (p. 360) shows that the bound $\lvert g(\alpha)\rvert<\lambda$ alone gives no free set of size $2$; the chapter does not mention the problem and decides nothing about it.

Results.

• Erdős-Tarski properties (pp. 348--349): each $P_m$ implies $P_{m+1}$ for $m=1,2,3$, $\neg P_1$ implies $\lambda$ strongly inaccessible, and for strongly inaccessible $\lambda$ each of $\neg P_1,\neg P_2,\neg Q,\neg R$ is equivalent to weak compactness.
• Erdős-Hajnal 1974 (p. 350): $P(\kappa,\kappa)$ holds if and only if $\kappa=\omega$ or $\kappa$ is weakly compact.
• Free Set Lemma (Hajnal, p. 359): if $\kappa<\lambda$ are infinite cardinals and $g$ maps $\lambda$ into $[\lambda]^{<\kappa}$, there is a free set of size $\lambda$; proved for regular $\lambda$ by Lázár (1936) and for singular $\lambda$ under GCH by Erdős (1950).
• Erdős 1954, Theorem 6 (pp. 359--360): if every $g(x)$ is nowhere dense there is a free set of size $\aleph_0$, everywhere dense by Bagemihl (1973); a free set of size $\aleph_1$ fails under CH for some such $g$, and every such $g$ has one of size $\aleph_2$ when CH fails and there is a Luzin set of size $\aleph_2$.
• Erdős-Hajnal-Máté 1973, Theorem 3.8 (p. 360): for regular $\lambda$ with Condition B holding for $\operatorname{ran}(g)$, there is a free set of size $\aleph_0$, one of size $\mu$ whenever $\mu<\lambda$ and $\nu^{<\mu}<\lambda$ for all $\nu<\lambda$, and one of size $\lambda$ if $\lambda$ is weakly compact.

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