A published paper or arXiv preprint answers the main question of Erdős Problem #550 (theorem erdos_550 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

Let m_1\leq\cdots\leq m_k and n be sufficiently large. If T is a tree on n vertices and G is the complete multipartite graph with vertex class sizes m_1,\ldots,m_k then prove that R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1. This problem is #16 in Ramsey Theory in the graphs problem collection. Li [Li26] proved this, combining an off-Turán tree-embedding theorem with a compactness theorem for bounded-rank hypergraph obstructions. The linked formal proof states the bound for trees on an arbitrary finite vertex type and with `K_{m_1,m_2}` written as a complete multipartite graph with two parts; the statement below is the special case `V = Fin n`.

References formal-conjectures lists for this problem (a starting point, not evidence):

• [Ch77] Chvátal, V., Tree-complete graph Ramsey numbers. J. Graph Theory (1977), 93.
• [Li26] Li, E., A Resolution of Erdős Problem 550 on Tree versus Complete Multipartite Ramsey Numbers, arXiv:2606.23659 (2026).