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)