**
Title:** Exact
Zarankiewicz
numbers
through
linear
hypergraphs

Speaker: | Daniel Horsley |

Affiliation: | Monash University |

Location: | Contact Sabrina Lato for Zoom link |

**
Abstract:** The
\emph{Zarankiewicz
number}
$Z_{2,2}(m,n)$
is
usually
defined
as
the
maximum
number
of
edges
in
a
bipartite
graph
with
parts
of
sizes
$m$
and
$n$
that
has
no
$K_{2,2}$
subgraph.
An
equivalent
definition
is
that
$Z_{2,2}(m,n)$
is
the
greatest
total
degree
of
a
linear
hypergraph
with
$m$
vertices
and
$n$
edges.
A
hypergraph
is
\emph{linear}
if
each
pair
of
vertices
appear
together
in
at
most
one
edge.
The
equivalence
of
the
two
definitions
can
be
seen
by
considering
the
bipartite
incidence
graph
of
the
linear
hypergraph.

A simple argument determines the exact value of $Z_{2,2}(m,n)$ whenever $n \geq \binom{m}{2}$. Guy was able to extend this determination to all $n \geq \binom{m}{2}/3+\frac{m+1}{3}$. This talk discusses how we can find, for large $m$, the exact value of $Z_{2,2}(m,n)$ in almost all of the remaining cases where $n=\Theta(m^2)$. Our results can be generalised to the Zarankiewicz numbers $Z_{2,\lambda+1}(m,n)$ by considering hypergraphs in which each pair of vertices appear together in at most $\lambda$ edges.