BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Perfect matchings in subgraphs of random hypergraphs Ferber, Asaf


Let $m_d(k,n)$ be the minimal $m$ such that every $k$-uniform hypergraph on $n$ vertices and with minimum $d$-degree at least $m$ contains a perfect matching (of course, assuming that $n$ is divisible by $k$). The problem of determining $m_{d}(k,n)$ has attracted a lot of attention in the last few decades, and unfortunately is still far from being completely understood. For example, already the cases $d=1$ and $k\geq 6$ are unknown. In this talk we show that random hypergraphs typically behave as expected. Specifically, we prove (without knowing $m_d(k,n)$) that for a typical $H^k_{n,p}$ with $p\geq n^{-c(d,k)}$, every spanning subgraph with minimum $d$-degree at least $(1+o(1))m_d(k,n)p$ contains a perfect matching. This is clearly (asymptotically) best possible. Joint work with Matthew Kwan.

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