Proof of the $(n/2-n/2-n/2)$ Conjecture for Large $n$

Abstract

A conjecture of Loebl, also known as the $(n/2 - n/2 - n/2)$ Conjecture, states that if $G$ is an $n$-vertex graph in which at least $n/2$ of the vertices have degree at least $n/2$, then $G$ contains all trees with at most $n/2$ edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi proved an approximate version of this conjecture. We prove it exactly for sufficiently large $n$. This immediately gives a tight upper bound for the Ramsey number of trees, and partially confirms a conjecture of Burr and Erdős.