Erdős #474 — 3-colouring $\\mathbb{R}^2$ vs. $2^{\\aleph_0}\\not\\to[\\aleph_1]_3^2$

Statement

Under what set-theoretic assumptions is it true that $\mathbb{R}^2$ can be $3$-coloured such that, for every uncountable $A\subseteq\mathbb{R}^2$, $A^2$ contains a pair of each colour?

Equivalently (as stated on erdosproblems.com/474): for which values of the continuum $\mathfrak c=2^{\aleph_0}$ is it true that $$2^{\aleph_0}\not\to[\aleph_1]_3^2\,?$$ (the negative square-bracket partition relation: a colouring $c:[\mathfrak c]^2\to 3$ such that every uncountable $U\subseteq\mathfrak c$ realizes all $3$ colours on its pairs).

Facts

- Prize \$100; erdosproblems.com/474 tags it "NOT PROVABLE" — its own special category meaning "open in general, but there exist models of set theory where the result is false" (i.e. genuinely a question of *which* set-theoretic universes satisfy it, not a single true/false fact). - Falsifiable: no — this is an independence-flavored question about ZFC-consistency across models, not resolvable by any finite computation. - Origin: a problem of Erdős from 1954, refs [Er95d, p.64] (Erdős, "On some problems in combinatorial set theory," Publ. Inst. Math. (Beograd) (1995), 61–65, MR1387354) and [Va99, 7.81] ("Some of Paul's favorite problems," booklet for the 1999 Budapest conference "Paul Erdős and his mathematics"). The same equation $2^{\aleph_0}\not\to[\aleph_1]_3^2$ is posed as problem 15(a) in Erdős–Hajnal's 1971 list [EH71] "Unsolved problems in set theory" — Shelah's 2026 paper (arXiv:2601.02923) opens by citing exactly this framing, so #474 is the same underlying question as EH71 #15(a). - Known partial results (per erdosproblems.com/474's own remarks, verified against its cited refs): - Sierpinski and Kurepa independently proved the 2-colour case is a ZFC theorem: $2^{\aleph_0}\not\to[\aleph_1]_2^2$ always holds, via an explicit colouring built from a well-ordering of the reals in order type $\mathfrak c$. - Erdős proved the 3-colour negative relation $2^{\aleph_0}\not\to[\aleph_1]_3^2$ holds under CH ($\mathfrak c=\aleph_1$). - Shelah [Sh88] ("Was Sierpinski right? I," Israel J. Math. 62 (1988), 355–380, MR955139) proved it is consistent without CH that the *positive* relation $2^{\aleph_0}\to[\aleph_1]_3^2$ holds — but only with a very large value of $\mathfrak c$ (the original construction needed the continuum to be an "ex-large cardinal," in the best case the first weakly Mahlo cardinal — confirmed via the abstract of the 2026 sequel, arXiv:2601.02923, which frames its own improvement against exactly this baseline). - The exact open remainder, per [Va99]: is it consistent to have the *negative* answer ($2^{\aleph_0}\not\to[\aleph_1]_3^2$) when $\mathfrak c=\aleph_2$ specifically? This is the $100 question and remains open as of this search (2026-07-02). - Related problems: none listed as explicit "See also" links on erdosproblems.com/474 itself (checked — the page has no cross-reference section, only the set theory | ramsey theory tags).

Literature state

Not resolved; genuinely active area with a very recent (Jan 2026) advance that narrows but does not close the gap.

- Saharon Shelah, "Consistency of square bracket partition relation," arXiv:2601.02923 (submitted 6 Jan 2026, this is publication #1258 on Shelah's list). Read in full (intro + theorem statements via pypdf text extraction). Explicitly frames itself as continuing the Erdős–Hajnal 15(a) program: "[Komjáth [Kom25]] provided a comprehensive update on this topic," and lists its own lineage as [She88 §2] (=[Sh88] on erdosproblems.com), [She92], [She89], [She95], [She96], [She00], and Rabus–Shelah [RS00]. Main new result (Thm 0.2, worked example Thm 0.1): builds a $(<\lambda)$-support iterated forcing giving positive relations $\theta\to[\partial]^2_{\sigma,2}$ with the continuum set to a *small, non-large* cardinal and *no large-cardinal hypothesis* — e.g. concretely $\mathfrak c=2^{\aleph_0}=\aleph_6$ forces $\aleph_5\to[\aleph_2]^2_{n,2}$ for all $n\geq3$. This is a genuine improvement in the "how small can $\mathfrak c$ be while forcing the positive relation" direction, but the indices in the worked example ($\aleph_2,\aleph_5,\aleph_6$) do not match the exact $(\aleph_1,\mathfrak c=\aleph_2)$ instance that #474 leaves open, and the paper does not claim to resolve that specific case — a follow-up "$[S^+]$ in preparation" is mentioned for the higher-arity ($n>2$ superscript) case only. So this is progress on the surrounding research program, not a resolution of the $100 question itself. - Péter Komjáth, "The Erdős–Hajnal Problem List," Bull. Symbolic Logic 31(3):418–461 (2025), DOI 10.1017/bsl.2025.1 — described in Shelah's 2026 paper as giving "a comprehensive update on this topic" for EH71 problem 15(a). Confirmed to exist via Crossref/Semantic Scholar metadata (DBLP: journals/bsl/Komjath25); full text is paywalled (Cambridge Core) and I could not verify whether it reports further progress specifically on the $\mathfrak c=\aleph_2$ sub-case — flagged as an unverified lead, worth chasing via institutional access. - Saharon Shelah, "Strong partition relations below the power set: consistency; was Sierpinski right, II?" arXiv:math/9201244 (1992) — directly germane to the "how small can the continuum be" boundary: disproves a related conjecture of Galvin that $2^\omega\geq\omega_2 \,\wedge\, \omega_2\to[\omega_1]^n_{h(n)}$ is consistent for suitable $h$ *exactly at* $\omega_2$ (§5, "we disprove this and give similar negative results"), while proving the positive-relation analogue *is* consistent when $\omega_2$ is replaced by the (large) continuum itself. This is the closest known technical precedent bearing on whether $\aleph_2$ specifically is an achievable threshold for these relations — it cuts in the direction of $\aleph_2$ being a genuinely hard boundary, though it does not directly address the negative-relation/$3$-colour instance of #474. - Shimon Garti, Yair Hayut, Saharon Shelah, "On a problem of Erdős and Hajnal," arXiv:2502.16625 (Feb 2025) — resolves a *sibling* Erdős–Hajnal question of the same shape (is local GCH indispensable for a negative ordinary partition relation, this time at $\lambda^+$ for $\lambda$ singular): shows the negative relation $\lambda^+\not\to(\lambda^+,(3)_\theta)^2$ remains consistent even when $2^\lambda>\lambda^+$ (GCH fails locally), using PCF theory. Structurally the same kind of question as #474's residual gap ("is [local instance of] GCH indispensable for this negative/positive relation?") solved via the modern PCF/Todorcevic toolkit — the strongest available *technique* analog even though it is not the same cardinal-arrow instance. - No AI-system contribution (LLM, AlphaEvolve, Aristotle, formal-conjectures) found for this problem in any source searched.

Attack surface

- Mode: literature-resolution + derivation (definitively not finite-search — this is a forcing/consistency question). - Concrete first experiment: not a runnable computational search. The one concrete, scoped task available is a derivation/formalization one: carefully instantiate Shelah's Theorem 0.2 machinery (arXiv:2601.02923, the $(<\lambda)$-support iteration with the "solution" structures $(U,\bar N,\bar\pi)$ of Def. 0.7/1.1–1.3) with the specific target indices $\sigma=3,\ \partial=\aleph_1,\ \mu=\aleph_2$ and check exactly where the hypothesis $\lambda=\lambda^{<\lambda}<\partial<\theta<\mu=\mu^\theta$ (Hyp. 1.1(a)) and the $2^{\partial+\ell}=\partial^{+\ell+1}$ side conditions in Thm 0.2 mechanically force $\mu\geq\aleph_6$-scale cardinals rather than $\aleph_2$ — i.e. locate precisely which structural constant in the construction is the obstruction to reaching $\mathfrak c=\aleph_2$, since that constant is the real target of any attack. - Oracle: none mechanical (this is a genuine mathematical/forcing question); the only oracle available is: does the (hypothetical) construction survive expert/Lean-formal scrutiny of the forcing iteration (no such formalization exists yet for this family — an actual research contribution, not a checkable artifact). - Feasibility: honest read — famous-and-hard, but with a live, recently-active research thread (a paper on this exact program was posted 6 Jan 2026). Not a target for a fast independent resolution; the realistic contribution is (a) obtaining and reading Komjáth's 2025 survey in full to see if it narrows the $\aleph_2$ case further than what's inferable from abstracts, and (b) tracking Shelah's promised follow-up "$[S^+]$" for the $n>2$ case, which may eventually touch the exact indices needed.

Related

- concept/square-bracket-partition-relations — the core object $\theta\to[\partial]^2_{\sigma,\kappa}$ (Def 0.3 in arXiv:2601.02923); #474 is the $(\theta,\partial,\sigma)=(2^{\aleph_0},\aleph_1,3)$ negative instance. - concept/lambda-support-iterated-forcing — the specific $(<\lambda)$-support iteration with elementary-submodel "solution" structures $(U,\bar N,\bar\pi)$ used by Shelah (arXiv:2601.02923, Def 0.7/1.2/1.3) to force positive square-bracket relations at small, non-large continuum without large cardinals. - concept/walks-on-ordinals — Todorcevic's minimal-walks/oscillation-mapping machinery (Todorcevic, "Walks on Ordinals and Their Characteristics," Birkhäuser 2007), the general ZFC toolkit for constructing negative square-bracket colourings (the 2-colour Sierpinski–Kurepa case is the prototype this generalizes). - concept/pcf-theory — Shelah's possible-cofinalities machinery underlying both the 2026 forcing construction and the Garti–Hayut–Shelah 2025 resolution (arXiv:2502.16625) of the sibling successor-of-singular Erdős–Hajnal question. - concept/continuum-hypothesis — CH resolves #474 outright (Erdős's original result); the entire remaining question is about the non-CH landscape and how small $\mathfrak c$ can be made. - solved/sierpinski-kurepa-well-ordering-coloring — the ZFC-provable 2-colour case, the seed technique (a strong colouring built directly from a well-order of $\mathbb R$) that CH-case and Shelah's forcing constructions both generalize. - solved/galvin-conjecture-disproof — Shelah, "Was Sierpinski right? II?" (arXiv:math/9201244): disproves Galvin's conjecture that the positive relation is consistent exactly at $\mathfrak c=\aleph_2$-scale (only at large continuum), the closest known technical precedent on the $\aleph_2$-threshold question. - solved/erdos-hajnal-successor-singular-partition — Garti, Hayut, Shelah, arXiv:2502.16625 (2025): resolves the sibling "is local GCH indispensable for this negative partition relation?" question at $\lambda^+$, $\lambda$ singular, via PCF theory — same question-shape as #474's residual gap, solved technique to port over. - Does the K3 partition-ordinal property imply the Kn property? (Erdős–Hajnal, disproved by Darby/Schipperus/Larson) — Schipperus/Darby: a *different* (ordinal, not cardinal) partition-calculus question in the same erdosproblems.com "set theory" cluster, PROVED via elementary-submodel/tree constructions — a broader analog showing the submodel-forcing toolkit closes adjacent partition problems. - erdos/1128 — Prikry–Mills (reported by Todorčević [To94], Komjáth [Ko25b]): DISPROVED sibling question on $\aleph_1^3$ colourings, same problem family (Erdős–Hajnal set-theoretic partition calculus).