Towards a hypergraph version of the Pósa-Seymour conjecture

One of the first results in graph theory was Dirac’s theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other problems and results all stating that in dense graphs we can find certain sparse spanning subgraphs. Along these lines, one of the most influential problems is the celebrated Pósa-Seymour conjecture. It claims, that for all $r\geq 3, n\geq r$, every $n$-vertex graph $G$ with minimum degree at least $(r- 1)n/r$ must contain the $(r-1)$-th power of a Hamilton cycle. This innocent-looking problem is still not known for all $n$ despite decades of study and dozens of papers.

In this paper, they authors make a significant step towards proving a hypergraph version of this difficult problem. More precisely, they show that for fixed $r \geq k \geq 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least

$$\left(1 - \left({r-1\choose k-1} + {r-2\choose k-2}\right)^{-1}\right)n + o(n)$$ contains the $(r - k + 1)$-th power of a tight Hamilton cycle. The result includes all previously known special cases. The proof follows from a more general embedding
theorem and uses the hypergraph regularity and absorption methods.