Abstract: For positive integers $d < k$ and $n$ divisible by $k$, let $m_{d}\left(k,n\right)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_{1}\left(2,n\right)=\lceil n/2 \rceil$. However, in general, our understanding of the values of $m_{d}\left(k,n\right)$ is still very limited, and it is an active topic of research to determine or approximate these values. In this talk we discuss a ``transference'' theorem for Dirac-type results relative to random hypergraphs. Specifically, for any $d< k$, and any "not too small" $p$, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and edge probability $p$ typically has the property that every spanning subgraph of $G$ with minimum $d$-degree at least $\left(1+o(1)\right)m_{d}\left(k,n\right)p$ has a perfect matching. Observe that this result holds even in cases that $m_{d}\left(k,n\right)$ is still unknown.
Joint work with Matthew Kwan.
Please contact Sean at SEnglish (at) illinois (dot) edu for the Zoom ID.