Li 2026 resolution erdos problem 550 tree versus

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theorem_1_1: The inequality of Problem 550 as stated by an unrefereed 2026 preprint that claims to prove it; recorded as an author's claim, not an accepted theorem.

theorem_3_2: The off-Turan tree-embedding theorem of an unrefereed 2026 preprint, a tool for its claimed proof of the Problem 550 inequality; recorded as an author's claim, statement checked, proof not read.

theorem_5_2: The finite null-blocker compactness theorem of an unrefereed 2026 preprint, a tool for its claimed proof of the Problem 550 inequality; recorded as an author's claim, statement checked, proof not read.

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E. Li, A resolution of Erdős Problem 550 on tree versus complete multipartite Ramsey numbers, arXiv:2606.23659 (v1 22 June 2026; v2 2 August 2026). Preprint; no journal record was found on 2026-09-17.

The copy read for this card is the arXiv v1 (stamped 22 Jun 2026, dated June 23, 2026), 20 pages numbered 1--20 and complete: Sections 1--9, the acknowledgments and the reference list are present, and every theorem, equation and section destination named inside it resolves within it, though the proofs cite several propositions, lemmas and Corollary 3.3 as "Theorem" (for example "Theorem 2.1" for Proposition 2.1, p. 5, and "Theorems 5.3 and 5.4" for Lemmas 5.3 and 5.4, p. 16). The arXiv listing's v2 (comment: "V2: The proof has been formally verified in Lean. 26 pages") was not compared; locators below are v1 pages. Source URL: https://arxiv.org/abs/2606.23659. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.23659), every other right reserved.

Read status: claims checked for Theorem 1.1, the lower bound (1) and the two-sided form (3) (read clause by clause on the page images of pp. 1--2), and for Theorems 3.2, 5.1 and 5.2 (pp. 4--5, 10--11 and 13); the proof (Sections 3--8) was not read, and its architecture is recorded from the paper's own summary on p. 2 and the section headings. The paper's claim to resolve the problem is recorded as a claim.

Contents

• Introduction (pp. 1--2): $R(J,L)$ and the complete multipartite graph $K_{m_1,\dots,m_k}$; Chvátal's $R(T,K_k)=(k-1)(|T|-1)+1$ for every tree [5]; Burr's canonical construction $R(J,L)\ge(\chi(L)-1)(n-1)+\sigma(L)$ for a connected $n$-vertex $J$ [4], giving (1) $R(T,F)\ge q(n-1)+a$ for $F=K_{m_1,\dots,m_k}$, $q=k-1$, $a=m_1$ (on $q(n-1)+a-1$ vertices, $q$ disjoint cliques of order $n-1$ and one of order $a-1$ in one color, every edge between them in the other); Erdős, Faudree, Rousseau and Schelp "proved the corresponding large-tree result when the smallest part has order 1" and asked the question (2) for arbitrary fixed part sizes, "question (2) of [7, p. 153]", recorded as Erdős Problem 550 (Li's [7] is the 1989 paper in Ann. New York Acad. Sci. 576).
• Theorem 1.1 (p. 2, claimed): fix $k\ge2$ and $1\le m_1\le\dots\le m_k$; there is $n_0=n_0(m_1,\dots,m_k)$ such that every $n$-vertex tree $T$ with $n\ge n_0$ satisfies $R(T,K_{m_1,\dots,m_k})\le(k-1)(R(T,K_{m_1,m_2})-1)+m_1$. With (1) this gives (3): $0\le R(T,F)-(q(n-1)+a)\le q(r-n)$, $r=R(T,K_{m_1,m_2})$.
• Proof architecture (p. 2 and the headings): the chain uniform EFRS asymptotic (Proposition 2.1, p. 3, "the uniform corollary following Theorem 2" of the 1985 paper [6]) ⟹ off-Turán embedding (Theorem 3.2, pp. 4--5, via Szemerédi regularity and the Hladký--Piguet regular-matching lemma, Section 3) ⟹ near-Turán red density (Corollary 3.3) ⟹ stable reservoirs (Erdős--Simonovits stability, Proposition 2.2; Lemma 6.1, Section 6) ⟹ profile and blocker inequalities (the profile Lemma 4.3 of Section 4 and Lemmas 7.1--7.2 of Section 7, "Blocker hypergraphs") ⟹ compactness rounding (the null-blocker theorems 5.1, pp. 10--11, and 5.2, p. 13, with shadow hypergraphs, Section 5) ⟹ Ramsey capacity contradiction (Section 8, p. 19). The proof of Theorem 1.1 itself runs through Sections 6--8 (pp. 16--19), with Sections 3--5 as its tools. Section 9 (p. 20): concluding remarks.
• Acknowledgments (p. 20): the author declares the use of OpenAI's ChatGPT "for ideation, formulation, proof exploration and refinement, narrowing the search space, programming, LaTeX formatting and other forms of orchestration" and takes "full responsibility for the accuracy of the final contents of this paper". Recorded as the source's own provenance declaration; no part of the argument was checked.
• Context cited (p. 2): Balla, Pokrovskiy and Sudakov (bounded-degree tree Ramsey goodness), Montgomery, Pavez-Signé and Yan (J. Combin. Theory Ser. B 173 (2025), bounded-degree trees versus general graphs) and Mi and Wang (arXiv:2605.26826, one large part).

Compiled scope

Pages 1--5, 9--11, 13, 16 and 18--20 were read on the page images, which carry every section heading, and the remaining statement labels in the text layer. The proof was not read, no step was checked, and nothing here is independently reviewed.

Bears on. #550: Theorem 1.1 states the problem's inequality (with $\chi(K_{m_1,\dots,m_k})=k$) for fixed part sizes and all sufficiently large trees; it is an unrefereed claim, and the page keeps the site's label. Theorems 3.2 and 5.2 bear on the problem only as steps of the claimed proof of Theorem 1.1.

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