\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Dhruv Mubayi and John Talbot}
%
%
\medskip
\noindent
%
%
{\bf Extremal Problems for $t$-Partite and $t$-Colorable Hypergraphs}
%
%
\vskip 5mm
\noindent
%
%
%
%
Fix integers $t \ge r \ge 2$ and an $r$-uniform hypergraph $F$. We
prove that the maximum number of edges in a $t$-partite $r$-uniform
hypergraph on $n$ vertices that contains no copy of $F$ is $c_{t,
F}{n \choose r} +o(n^r)$, where $c_{t, F}$ can be determined by a
finite computation.

We explicitly define a sequence $F_1, F_2, \ldots$ of $r$-uniform
hypergraphs, and prove that the maximum number of edges in a
$t$-chromatic $r$-uniform hypergraph on $n$ vertices containing no
copy of $F_i$ is $\alpha_{t,r,i}{n \choose r} +o(n^r)$, where
$\alpha_{t,r,i}$ can be determined by a finite computation for each
$i\ge 1$. In several cases, $\alpha_{t,r,i}$ is irrational. The main
tool used in the proofs is the Lagrangian of a hypergraph.

\bye