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Claim. For every $\gamma>0$ there is $n_0$ such that for all $n\ge n_0$ and all $k\ge\gamma n$, every graph on $n$ vertices with more than $\frac{k-1}2n$ edges contains every tree on $k+1$ vertices. The preprint states the conjecture with the tree's order as its parameter, more than $(k-2)n/2$ edges forcing every $k$-vertex tree, which is the site's statement with $k$ shifted by one; its abstract also derives, as a corollary, a solution of a problem of Erdős and Graham on the multicolor Ramsey numbers of trees.

Covers. The instances with $k$ at least a fixed positive fraction of $n$ and $n$ large in terms of that fraction, infinitely many for every $\gamma$. The instances with $k=o(n)$, and every instance with $n$ below the unstated $n_0$, are outside this claim; the full statement is settled by the accepted claim page [[problems/extremal_graph_theory/E0548/claims/2026_09_03_adamczewski|Adamczewski 2026]].

Depends on. Nothing in this wiki.

Standing. Claimed, not accepted. Bruce Reed and Maya Stein, The Erdős--Sós conjecture in dense graphs, arXiv:2609.05417 (the library card pages Theorem 2 and Corollary 4), v1 4 September 2026 (the date this page is named by), v2 8 September 2026, under the CC BY 4.0 license; no journal version is known. The site's commentary, last edited 7 September 2026, records the result as [ReSt26], but the site's label credits GPT-6 Astra with the full proof and is not an acceptance of this paper; nothing is refereed or formalized. The companion preprint of the same authors, The extremal cases of the Erdős--Sós conjecture (arXiv:2609.05411, [ReSt26b] on the site), proves the conjecture for host graphs with the minimum edge count that contain a subgraph of minimum degree at least $(1-\mu)k$; it restricts the host graph, settles no instance of the statement, and has no page.

Read depth. The arXiv abstract and version record were read; the paper's body was not read, and nothing is independently reviewed in this corpus.