A short proof of the Erd\H os--S\'os Conjecture

Abstract

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 a substantially simplified proof of an analogous result for eulerian digraphs, again proved by GPT-6 Astra, and announced by Mubayi and Verstraete. In addition, we determine the extremal (di)graphs for all three problems.