Lee: Erdős Problem 623 and the free-subset property

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corollary_5_1: In ZFC the positive answer to Problem 623 is equivalent to the free-set properties FS_1 and FS_omega at (aleph_omega, omega) and to Koepke's free-subset property Fr_omega(aleph_omega, omega).

proposition_2_2: In ZFC the positive answer to Problem 623 is equivalent to FS_1(aleph_omega, omega), that every map sending finite subsets of aleph_omega to sets of at most one point has a countably infinite free set.

proposition_3_1: In ZFC, for every infinite cardinal kappa, FS_1(kappa, omega) is equivalent to FS_omega(kappa, omega): free sets for maps with forbidden sets of size at most one give free sets for countable forbidden sets.

proposition_4_3: In ZFC, for all infinite cardinals kappa, mu and lambda, the free-set property FS_mu(kappa, lambda) for maps on finite sets is equivalent to Koepke's free-subset property Fr_mu(kappa, lambda) for structures.

theorem_1_1: Lee's main theorem: ZFC plus the positive answer to Problem 623 is consistent exactly when ZFC plus a measurable cardinal is, and ZFC plus the negative answer is consistent exactly when ZFC is.

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Sungchul Lee, Erdős Problem 623 and the Free-Subset Property. preprint (GitHub, June 2026). No copyright or license line is printed on the six pages (first and last read in full); the source repository shows no LICENSE file and no license badge, and its README states no terms (https://github.com/lsngchl/Erdos623); the term is unstated.

Theorem 1.1 states that ZFC + E623 is consistent if and only if ZFC plus a measurable cardinal is consistent, while ZFC + not-E623 is consistent if and only if ZFC is consistent. The proof establishes in ZFC (Corollary 5.1) the chain E623 <=> FS_1(aleph_omega, omega) <=> FS_omega(aleph_omega, omega) <=> Fr_omega(aleph_omega, omega). Here FS_mu(kappa, lambda) says that every map F from the finite subsets of kappa to subsets of kappa of size at most mu has an F-free set Y of size lambda, one with F(A) disjoint from Y \ A for every finite A in Y (Definition 2.1), and Fr_mu(kappa, lambda), Koepke's free-subset property, says that any structure with at most mu functions and relations, all ordinals below kappa lying in its universe, has a free set X contained in kappa with |X| >= lambda (Definition 4.2). Proposition 2.2 restates the problem (a function f on finite subsets of a set of size aleph_omega with f(A) not in A, seeking an infinite independent Y) as FS_1(aleph_omega, omega), Proposition 3.1 upgrades singleton-valued to countable-valued forbidden sets by encoding the enumeration of F(A) through an injection N from omega x omega to omega with N(i,k) > k (for example N(i,k) = 2^i(2k+1)) on the initial segments of a candidate free set, and Proposition 4.3 identifies FS_mu(kappa, lambda) with Fr_mu(kappa, lambda) for infinite kappa, mu and lambda. Koepke's theorems, recalled as Theorem 5.2 (Fr_omega(aleph_omega, omega) gives an inner model with a measurable cardinal at most aleph_omega, and a measurable cardinal gives a two-stage generic extension in which Fr_omega(aleph_omega, omega) holds), then give the first equivalence of Theorem 1.1 (Corollary 5.3); the first of them, with Scott's theorem that a measurable cardinal implies V != L, shows that Fr_omega(aleph_omega, omega) and hence E623 fail in L, which gives the second (Corollary 5.4). For problem 623 the preprint claims independence from ZFC, relative to the consistency of a measurable cardinal, with the positive side having measurable-cardinal strength, which would confirm Erdős's own suggestion that the aleph_omega case might be undecidable.

Source: https://github.com/lsngchl/Erdos623.

Bears on. #623: Theorem 1.1 states that ZFC plus a positive answer is consistent if and only if ZFC plus a measurable cardinal is, and that ZFC plus a negative answer is consistent if and only if ZFC is; Corollary 5.1 states in ZFC that the positive answer is equivalent to Koepke's Fr_omega(aleph_omega, omega). The preprint is unrefereed, and the result is recorded as a claim on Lee's claim page.

Results.

• Theorem 1.1 (p. 1; proved as Corollaries 5.3, p. 5, and 5.4, p. 6): ZFC + E623 is equiconsistent with ZFC plus a measurable cardinal; ZFC + not-E623 is equiconsistent with ZFC.
• Proposition 2.2 (p. 2, with Definition 2.1): in ZFC, E623 holds if and only if FS_1(aleph_omega, omega) holds.
• Proposition 3.1 (p. 2; proof pp. 2--3): in ZFC, for every infinite cardinal kappa, FS_1(kappa, omega) is equivalent to FS_omega(kappa, omega).
• Proposition 4.3 (p. 4, with Definitions 4.1, p. 3, and 4.2, p. 4; proof pp. 4--5): in ZFC, for all infinite cardinals kappa, mu, lambda, FS_mu(kappa, lambda) is equivalent to Fr_mu(kappa, lambda).
• Corollary 5.1 (p. 5): in ZFC, E623 <=> FS_1(aleph_omega, omega) <=> FS_omega(aleph_omega, omega) <=> Fr_omega(aleph_omega, omega).

Read status: claims checked for the statements above, read clause by clause on the printed pages; the proofs were read but not checked.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.