This paper settles a long-standing question of Paul Erdős about Ramsey numbers that compare trees with complete multipartite graphs. A Ramsey number R(J,L) is the smallest number N such that every red–blue coloring of the edges of the complete graph on N vertices contains either a red copy of J or a blue copy of L. The question, known as Erdős Problem 550, asked for a sharp upper bound on R(T, K_{m1,…,mk}) when T is any n-vertex tree and the k-part multipartite graph K_{m1,…,mk} has fixed part sizes m1 ≤ … ≤ mk.

The main theorem gives a clean reduction from the general multipartite case to a simpler bipartite case. For fixed k ≥ 2 and fixed part sizes m1 ≤ … ≤ mk, the authors prove that for every sufficiently large n and for every n-vertex tree T, R(T, K_{m1,…,mk}) ≤ (k−1) (R(T, K_{m1,m2}) − 1) + m1. In words: the Ramsey number against the k-part graph is bounded by (k−1) times the Ramsey number against the two smallest parts, plus a small offset depending on m1. They also show a quantitative reformulation that the excess above the obvious lower bound is controlled by the excess in the bipartite problem involving the two smallest parts.

The proof combines several modern tools from extremal graph theory. One key new ingredient is an off-Turán tree-embedding theorem, which lets the authors embed large trees into graphs whose edge density is just above the Turán threshold (a standard density threshold for forbidding a fixed graph). To build that embedding they use Szemerédi’s regularity lemma (a way to approximate large graphs by a bounded number of random-like pieces) and a local regular-matching embedding lemma of Hladký and Piguet. The other main component is a compactness-and-rounding theorem for certain bounded-rank hypergraph obstructions; this relies on the idea of shadow hypergraphs to keep track of obstructions that might “escape” along a limiting sequence.