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arXiv · 2606.23659

A Resolution of Erd\H{o}s Problem 550 on Tree versus Complete Multipartite Ramsey Numbers

Eric Li
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Abstract

We resolve Erd\H{o}s Problem 550, originally asked as question (2) of Erd\H{o}s, Faudree, Rousseau, and Schelp. Precisely, for fixed integers $k\geq 2$ and $1\leq m_1\leq \cdots \leq m_k$, we prove that, for every sufficiently large $n$ and every $n$-vertex tree $T$, $R(T,K_{m_1,\ldots,m_k}) \leq (k-1)(R(T,K_{m_1,m_2})-1)+m_1$. The proof combines an off-Tur\'an tree-embedding theorem, proved by regularity and whole-edge allocation, with a compactness theorem for bounded-rank hypergraph obstructions. The full and unconditional proof has been formally verified in Lean.

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Eric Li. 2026-06-22. A Resolution of Erd\H{o}s Problem 550 on Tree versus Complete Multipartite Ramsey Numbers. https://arxiv.org/abs/2606.23659

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