Loebl–Komlós–Sós conjecture for trees

Let $G$ be an $n$-vertex graph, let $d$ be a nonnegative integer, and let $T$ be a tree with $d$ edges.

Loebl–Komlós–Sós conjecture. If at least $n/2$ vertices of $G$ have degree at least $d$, then $G$ contains a copy of $T$.

The conjecture replaces the average-degree condition in the Erdős–Sós conjecture by a median-degree condition. The source notes that it has been proved in several important cases, including large dense host graphs, while the general sparse bounded-degree form remains an active problem.