The Erd\H{o}s--S\'os Conjecture: A visual exposition of the proof discovered by GPT-6 Astra

Abstract

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 the counting objects as \emph{early neighbors} in a reveal-and-stop procedure, organize the induction through explicit partitions and reversible swaps, and develop the proof through worked examples with consistent drawings. We also give direct and probabilistic conclusions, compare this formulation with other recent expositions, and record the classical consequence $R(T;q)\le q(t-2)+2$ for multicolor Ramsey numbers of trees.