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The case $r=n$ is considered in this paper by Martin and Wong. They prove that for every $n \geq 2$ and every $\epsilon >0$, the probability that a random $n \times n$ matrix with entries from $\{-k, …

Not an answer, but you may also be interested in the 'finite field' analogue of your question. Namely, given a finite field $\mathbb{F}_q$, what is the maximum size of a subset of vectors $S \subset …

Theorem. Let $A$ and $B$ be families of subsets of $[n]$, such that for all $a \in A$ and $b \in B$, $|a \cap b|$ is odd. Then $|A||B| \leq 2^{n-1}$. I will present two proofs of this theorem. One …

Perhaps I misunderstand the question, but isn't $S$={$1,2, \ldots n$} a counterexample? Thanks for the comments below. Edit: I still haven't solved the problem, but I managed to translate the prob …

No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfy your property are called base orderable matroids. There are …

Up to simplification (suppressing loops and parallel elements), every rank two matroid is just a rank two uniform matroid. Note that the vectors $(1, a_1), \dots, (1, a_n)$ represent the uniform ma …

Here is a proof that it is always possible by keeping at most $n+1$ equations throughout. Suppose the system $Ax=b$ has a unique solution $c \in \mathbb{Q}_{\geq 0}^n$, where $A \in \mathbb{Z}_{\geq 0 …

Terry Tao has a nice blog post on a 'cheap version' of the Kabatjanskii-Levenstein bound mentioned in Lucia's answer, using the so-called 'tensor product trick'.

This can be done via linear programming. Consider a set of linear inequalities $Ax \leq b$, together with an additional inequality $c^Tx \leq d$. We wish to know if the constraint $c^Tx \leq d$ is r …

The problem of determining a minimum support solution of a linear system is indeed NP-hard. Here is a reduction that works over the binary field $\mathbb{F}_2$. Given a graph $G$, an odd dominating …

It is not true that $\text{rank}_+^*(V) \leq \text{rank}_+(V) $. In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily colum …

This does not technically answer your question, but I think it may of interest to you, so bear with me. If you are interested in excluded-minor characterizations for real-representability, the situat …

No, this is not true. Let $G$ be the bowtie graph (this is the graph obtained by gluing two triangles at a vertex $u$). Then, $G$ does not have a spanning regular subgraph, but $\mathbb{1}$ is in th …

I will turn my comment above into a self-contained answer. Given a hypergraph $H=(V,E)$ and $X \subseteq V$, we say that $X$ is shattered if for all $X' \subseteq X$, there exists $e \in E$ such that …

The dimension of the circuit space of a matroid $M$ is the corank of $M$ if and only if $M$ is binary. Here is a proof. Given a basis $B$ and $e \notin B$, we let $C(e,B)$ be the unique circuit conta …