The extremal conjecture for tight paths and cycles in uniform hypergraphs

Let $k$-graphs be $k$-uniform hypergraphs, and let $n$ and $\ell$ be positive integers. A tight path on $\ell$ vertices is a sequence of $\ell$ distinct vertices in which every set of $k$ consecutive vertices forms an edge; a tight cycle is defined cyclically in the analogous way.

Extremal conjecture for tight paths and cycles. For any $\ell$, every $n$-vertex $k$-graph with more than

$$\frac{\ell-k}{k}\binom{n}{k-1}$$

edges contains a tight path on $\ell$ vertices, and every $n$-vertex $k$-graph with more than

$$\frac{\ell-1}{k}\binom{n-1}{k-1}$$

edges contains a tight cycle of length at least $\ell$.

This is proposed as the exact analogue for tight paths and cycles of the Erdős–Gallai theorem, complementing the paper's approximate extremal result. The conjecture concerns the sharp edge thresholds for forcing these structures; its resolution is not specified in the source.