Georgia Institute of Technology College of Sciences

For positive integers $r > \ell \geq 1$, an $\ell$-cycle in an $r$-uniform hypergraph is a cycle where each edge consists of $r$ vertices and each pair of consecutive edges intersect in $\ell$ vertices. For $\ell \geq 2$, we determine the exact threshold for the appearance of Hamilton $\ell$-cycles in an Erd\H{o}s--R\'enyi random hypergraph, confirming a conjecture of Narayanan and Schacht. The main difficulty is that the second moment is not tight for these structures. I’ll discuss how a variant of small subgraph conditioning and a subsampling procedure overcome this difficulty.