Pósa–Seymour conjecture for powers of Hamilton cycles

Let $G$ be a graph on $n\geq k\geq 2$ vertices, and let $\delta(G)$ denote its minimum degree. The Pósa–Seymour conjecture. If

$$\delta(G) \geq \frac{k}{k+1}n,$$

then $G$ contains the $k$th power of a Hamilton cycle.

This conjecture determines the minimum degree threshold for forcing powers of Hamilton cycles in graphs. The source notes that Komlós, Sárközy and Szemerédi proved it for sufficiently large graphs; the stated finite formulation is therefore solved in the asymptotic sense described there.