Graph Theory seminar

A Proof of the Manickam-Mikl\'{o}s-Singhi Conjecture for Vector Spaces

Abstract:

Let $V$ be an $n$-dimensional vector space over a finite field.
Assign a real-valued weight to each $1$-dimensional subspace in $V$ so
that the sum of all weights is zero. Define the weight of a subspace $S
\subset V$ to be the sum of the weights of all the $1$-dimensional
subspaces it contains. We prove that if $n \geq 3k$, then the number of
$k$-dimensional subspaces in $V$ with nonnegative weight is at least the
number of $k$-dimensional subspaces in $V$ that contain a fixed
$1$-dimensional subspace. This result verifies a conjecture of Manickam
and Singhi from 1988.