Erdős–Sós conjecture for trees

Prove that for every connected tree T with k edges, the extremal number satisfies ex(n,T)=(k-1)n/2.

Background

The classical extremal number ex(n,T) is the maximum number of edges in an n-vertex graph containing no copy of a forbidden tree T. For a k-edge tree, the Erdős–Stone–Simonovits theorem gives only the asymptotic bound ex(n,T)=o(n2). The Erdős–Sós conjecture predicts the exact linear extremal threshold and asserts that it depends only on the number of edges of T, not on its structure.

The paper notes that the conjecture is known for several families, including paths, caterpillars, spiders of diameter at most four, and trees with sufficiently many leaves adjacent to one vertex, but remains unresolved in general.

References

The following Conjecture of Erdős and Sós concerning $ex(n,T)$ is one of the most well-known in extremal graph theory. For every $k$-edge connected tree $T$ we have $ex(n,T)={\frac{(k-1)}{2}n}$.

— Rainbow Erdős-Sós Conjectures  (2502.00135 - Crawford et al., 31 Jan 2025) in Conjecture 1, Section 1.1, 'The Erdős–Sós Conjecture'

In the graph case, the Erd˝os-S´os conjecture [8] states that ex(n, T ) ≤ (t − 1)n/2.

— On Turán problems for suspension hypergraphs  (2502.10905 - Cheng et al., 15 Feb 2025) in Section 1, p. 2

Conjecture 1 (Erdős-Sós Conjecture) For any tree T, ex(n, T) ≤ |T|−2/2 n. Here, |T| denotes the number of vertices in T. Since the conjecture was first proposed in 1962, a lot of efforts have been made, but it has not been solved.

— Turán problems for suspension of a balanced tree  (2503.05166 - Zhu et al., 7 Mar 2025) in Section 1, immediately following Conjecture 1 (page 4)

The Erdős--Gallai theorem shows that the conjecture is true for a path. Ajtai, Komlós and Szemerédi announced a proof of the Erdős--Sós conjecture for large $t$, and notable works (for example, see ) have verified the Erdős--Sós conjecture in various special cases, but the conjecture remains widely open.

— Large Cliques and Clique Spectral Radius in the Erdős--Sós Problem  (2608.25746 - Zhao et al., 26 Aug 2026) in Section 1, Introduction

The following famous conjecture has been open for over sixty years.

\begin{conjecture}[Erd\H os--S os conjecture]\label{es} For $k, n\in\mathbb N+$, every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree as a subgraph. \end{conjecture}

— The Erd\H os-Sós conjecture in dense graphs  (2609.05417 - Reed et al., 4 Sep 2026) in Section 1, Introduction; Conjecture 1

The following famous conjecture was made in the early 1960s (see). For $k, n\in\mathbb N$, every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree.

— The extremal cases of the Erd\H os--Sós conjecture  (2609.05411 - Reed et al., 4 Sep 2026) in Introduction, Conjecture 1 (labelled Conjecture~\ref{es})