The additive basis conjecture

Importance: High ✭✭✭
Subject: Combinatorics
» Matrices
Recomm. for undergrads: no
Posted by: mdevos
on: March 8th, 2007

\begin{conjecture} For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\mathbb Z}_p)^n$ contains an additive basis. \end{conjecture}

Definition: Let $V$ be a finite dimensional vector space over the field ${\mathbb Z}_p$. We call a multiset $B$ with elements in $V$ an additive basis if for every $v \in V$, there is a subset of $B$ which sums to $v$.

It is worth noting that this conjecture would also imply that every $2c(3)$-edge-connected graph has a nowhere-zero 3-flow, thus resolving \OPref[The weak 3-flow conjecture]{3_flow_conjecture}.