Erd\H{o}s-S\'{o}s via random cyclic orderings

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. 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.