Erdős Problem #548: the Erdős–Sós conjecture

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

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erdosproblems.com/548: status and Thomas Bloom's proof exposition evidence for 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. Independent human corroboration, September 2026: Riordan and Scott gave a short proof of the conjecture (arXiv:2609.15893) and Wood published an expository account of the argument (arXiv:2609.17877). Neither used AI. Mubayi and Verstraëte then extended the method, with GPT-6 Astra, to Kalai's 1984 hypergraph conjecture (arXiv:2609.08012) and to a sharp directed analogue (arXiv:2609.10987), both catalogued separately. parent of this event

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

This event attributed to Tom Adamczewski

Solution.lean and the Lean development evidence for this event

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

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