A Resolution of Erdős Problem 550 on Tree versus Complete Multipartite Ramsey Numbers

Abstract

We resolve Erdős Problem 550, originally asked as question (2) of Erdős, 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 a new off-Turán tree-embedding theorem with a compactness-and-rounding theorem for represented bounded-rank hypergraph obstructions. The embedding theorem follows from Szemerédi regularity and a local regular-matching embedding lemma of Hladký and Piguet. The compactness argument uses shadow hypergraphs to retain obstructions whose vertices escape along the limiting sequence.