A simplified proof of the Erd\H{o}s-S-S\'os conjecture on trees in graphs is presented, which is based on its recent resolution by GPT-6 Astra and characterized the extremal examples for Kalai's conjecture.
Abstract
We present a simplified proof of the Erd\H{o}s-S\'os conjecture on trees in graphs, which is based on its recent resolution by GPT-6 Astra. At the same time, we also give another proof of the corresponding conjecture for antidirected trees in digraphs, posed by Addario-Berry, Havet, Linhares Sales, Reed, and Thomass\'e, and we simplify recent proofs of Kalai's conjecture for tight trees in hypergraphs. Our proof uses the language of random cyclic orderings in order to take advantage of symmetries that were obscured in the original argument, and we hope that the reader will find it intuitive. We also characterize the extremal examples for Kalai's conjecture, and we generalize it to the setting of so-called fully directed $r$-uniform hypergraphs, whose edges are ordered $r$-tuples.
The Erd\H{o}s--S\'os theorem states that every graph of average degree greater than $t-2$ contains every tree on $t$ vertices. A short counting proof was discovered by GPT-6 Astra in 2026. We give a visual, reader-centered exposition of that argument whose presentation differs substantially from the original. We recast...
It is shown that every Eulerian digraph on $n$ vertices with more than $(t-1)n$ arcs contains every oriented tree with $t$ edges. The digraphs have no loops or repeated arcs, but opposite arcs are permitted. The bound is sharp for each fixed oriented tree, as witnessed by disjoint unions of complete bidirected graphs....
The Erd\H os--S\'os Conjecture was recently proved by GPT-6 Astra, using a very ingenious and surprising argument. In this note, we present a simplified version of this argument in an (arguably) more natural form. We also prove a related conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomass\'e, and give...
The Erd\H{o}s-Gy\'arf\'as conjecture posits that every graph with minimum degree at least three contains a cycle of length some power of two. We prove a few simple structural properties for any minimal counter-example to this conjecture. In particular, the fraction of its vertices of degree three must be greater than $...
The celebrated conjecture of Lov\'asz from 1969 asks whether every connected vertex-transitive graph has a Hamiltonian path. Buci\'c, Christoph, Pokrovskiy and Steiner recently proved that every such graph on $n$ vertices contains a cycle of length $n^{2/3-o(1)}$. In this paper, we improve this bound to $n^{1-o(1)}$. O...
In 1977, Erd\H{o}s posed the problem of determining the maximum number $f_r(n)$ of edges in an $n$-vertex $r$-uniform hypergraph in which all disjoint pairs of edges have distinct unions. F\"uredi later conjectured that, for every fixed $r\ge 4$ and all sufficiently large $n$, $f_r(n)=\binom{n-1}{r-1}+\lfloor \frac{n-1...
Hao-Wei Huang, Jie Ma, Tian-Chi Yang· 2 citations· ⚡1
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