The Erdős–Sós conjecture predicts a simple connection between how many edges a large graph has and which trees it must contain. It says that any n-vertex graph with more than (k−2)n/2 edges must contain every tree with k vertices. The authors prove this statement when the tree is not too small: for every fixed fraction γ>0 there is a threshold n0 so that whenever n≥n0 and k≥γn the conjecture holds. As a corollary they solve a 51-year-old question of Erdős and Graham about multicolour Ramsey numbers of trees.

The conjecture is best possible in simple cases. For example, a complete graph on k−1 vertices has many edges but contains no k-vertex tree. That is why the conjectured edge bound is sharp. The new result shows that if the tree’s size k grows linearly with the size of the host graph, then the conjectured bound indeed forces every k-vertex tree to appear, once the host graph is large enough.

To prove this, the authors argue by contradiction. They assume there is a smallest counterexample and work with a “robust” version of it — a graph whose proper subgraphs all have smaller average degree. They cut the target tree into a small central piece plus many small rooted components. Then they apply Szemerédi’s Regularity Lemma, a standard tool that approximates a large graph by a simpler “reduced” graph made of a bounded number of clusters. This lets them plan where each small piece of the tree should go by matching those pieces to edges between clusters in the reduced graph.

A key part of the work is to find useful structure inside the reduced graph. The dense host guarantees clusters with large degree, but some neighbourhoods can be dead ends for embedding. The authors find either two adjacent clusters that are well connected, or two matchings of cluster pairs that cover much of the neighbourhood, and they record the obstructions when these covers fail. They then run careful, case-by-case embedding strategies. In some cases they balance how much of each bipartition class of the small tree parts goes to each side of a cluster pair. In other cases they embed some roots “invertedly” or use a specially chosen cluster with powerful neighbours to route around bottlenecks. The embedding steps are coordinated and sometimes delicate; timing and order matter.