# Ryser's conjecture for multipartite hypergraphs

###
Ian Wanless

Monash University

PDF

**Minisymposium:**
COMBINATORICS

**Content:**
An {\em $r$-partite hypergraph} has a partition of the vertices
into disjoint sets $V_1,\dots,V_r$ and every edge includes
exactly one vertex from each $V_i$. A {\em matching} in a hypergraph $H$ is
a set of disjoint edges. The largest size of a matching in $H$ is
denoted $\nu(H)$. A {\em cover} in a hypergraph is a set of vertices which
meets every edge. The size of the largest cover in $H$ is denoted $\tau(H)$.
Ryser conjectured that $\tau(H)\le(r-1)\nu(H)$ for $r$-partite hypergraphs.
Define $f(r)$ to be the smallest number of edges in an $r$-partite
hypergraph $H$ with $\nu(H)=1$ and $\tau(H)\ge r-1$ (if such an $H$ exists!).
I will discuss work from two recent papers in which we
\begin{itemize}
\item Prove the special case of Ryser's conjecture where $r\le9$ and $H$
is linear (meaning each pair of edges meets at a unique vertex).
\item Disprove a strengthened version of Ryser's conjecture proposed
by Aharoni.
\item Find the values of $f(6)$ and
$f(7)$ and improve the general lower bound on $f(r)$.
\end{itemize}