Erdős Problem #548: the Erdős–Sós conjecture

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Erdős Problem #548: the Erdős–Sós conjecture parent of this event

For every $n,k$ with $k+1\le n$, every simple graph $G$ on $n$ vertices satisfying $$|E(G)|\ge \frac{k-1}{2}n+1$$ contains every tree on $k+1$ vertices. The proof counts pairs $(\pi,j)$ where $\pi=(v_1,\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly $$2|E(G)|(n-1)!$$ such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus $$(k-1)n!.$$ If the target tree is absent, the rooted-copy term vanishes and one obtains $$2|E(G)|\le (k-1)n,$$ contradicting the density hypothesis. parent of this event

VibeMathed record: Erdős Problem #548: the Erdős–Sós conjecture evidence for this event

Erdős Problem #548: the Erdős–Sós conjecture evidence for this event

erdosproblems.com/548: status and Thomas Bloom's proof exposition evidence for this event

Challenge.lean: the compared statement evidence for this event

Solution.lean and the Lean development evidence for this event

Epoch AI, Announcing FrontierMath Erdős (1 September 2026) evidence for this event

This event attributed to GPT-6 Astra (pre-release)

This event attributed to Tom Adamczewski