\magnification=1400\nopagenumbers\hsize=6truein\noindent{\bf
Anant P.~Godbole, Ben Lamorte, Erik Jonathan Sandquist\smallskip\noindent
 Threshold Functions for the Bipartite Tur\'an Property}
\vskip1cm\noindent
Let $G_2(n)$ denote a bipartite graph with $n$ vertices in each color class,
 and let $z(n,t)$ be the bipartite Tur\'an number, representing the maximum
possible 
 number of edges in $G_2(n)$ if it does not contain a copy of the complete
 bipartite subgraph $K(t,t)$.  It is then clear that $\zeta(n,t)=n^2-z(n,t)$
 denotes the minimum number of zeros in an $n\times n$ zero-one matrix that
 does not contain a $t\times t$ submatrix consisting of all ones.  We are
 interested in the behaviour of $z(n,t)$ when both $t$ and $n$ go to
 infinity.  The case $2\le t\ll n^{1/5}$ has been treated elsewhere; here we use
 a different method to consider the overlapping case $\log n\ll t\ll n^{1/3}$.
  Fill an $n \times n$ matrix randomly
 with $z$ ones and $\zeta=n^2-z$ zeros.  Then, we prove
 that the
 asymptotic probability that there are no $t \times t$ submatrices with all
 ones is zero or one, according as
$z\ge(t/ne)^{2/t}\exp\{a_n/t^2\}$
or
$z\le(t/ne)^{2/t}\exp\{(\log t-b_n)/t^2\}$, where $a_n$ tends to
infinity at a specified rate, and $b_n\to\infty$ is arbitrary.
The proof employs the extended Janson exponential inequalities.
\bye