Seminar at SMU Delhi
December 14, 2012 (Friday) ,
3:30 PM at Webinar
Caltech (California Institute of Technology), USA.
A zero-sum Ramsey-type problem for hypergraphs and diagonal (Smith) forms of certain incidence matrices.
Abstract of Talk
The Smith normal form (or a diagonal form) of
an integer matrix $A$ allows us to easily determine whether
an a system $Ax=b$ of equations has an integer solution $x$.
We determine a diagonal form for a matrix $N=N_t(H)$
associated with a $t$-uniform hypergraph whenever $H$ has a
certain property that we call `primitivity', a previously
unstudied property that we show holds almost surely for a
random hypergraphs as the number of vertices increases. The
binary case of a zero-sum Ramsey-type problem of Alon and Caro
is equivalent to solving the congruences $Nx\equiv j\pmod 2$
where $j$ is the vector of all 1's, and so we can solve their
problem for primitive hypergraphs. This is joint work with