# Tagged Questions

Matroids are a common generalization of linearly independent sets and independent sets in graphs. Among other applications, they are exactly the simplical complexes in which the greedy algorithm outputs the optimal solution. Matroids are also studied for their own sake.

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### Multi cover a matroid with circuits

The following question looks similar to the double cover conjecture for graphs and regular matroids, but I am not sure if it has been studied for general matroids: Is there a universal constant $c$ ...
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### Are Oriented Graphs Related to Oriented Matroids?

My professor said that oriented matroids make it easier to investigate things such as connectivity. Recall that an oriented graph is a digraph without multiple edges or loops. Now Are oriented graphs ...
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### Introductory text on partitions, matroids, geometric lattices

Can anyone recommend a text which explains matroids, lattices of subsets, and how they are related? Possibly motivated with examples from different applications or areas of math.
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### Finite prime field representation of uniform matroid $U_{2,n}$

Suppose I have a uniform matroid $U_{2,n} = (E, I)$ (so $F \subset E$ has $F \in I \iff |F| \leq 2$) and want to represent it over $GF(p)$, i.e. I would like to construct a map $\phi : E \to GF(p)^2$ ...
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### The subgroup of $PGL(V)$ stabilizing a projective configuration

Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the ...
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### How to maximize this set function!?

Given a set $F$ and a function $p: 2^F \times 2^F \to [-1,0]$ such that $p (A \cup B, C) \leq p (A,C)$ for any sets $A, B, C \in 2^F$ : Q1: How can we choose a non-empty set $O \in 2^F$ such ...
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### Number of Different Bases in a Matroid

I am trying to figure out a lower bound on the number of different bases a matroid can have. The matroid has rank n and its ground set is the disjoint union of two bases. I think that it's $2^n$ ...
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### the rank of a matroid $\mid X \mid + r(E\backslash X)-r(E) \geq 0$

If I have the matroid $M=(E,I)$ and $\mid X \mid + r(E\backslash X)-r(E) \geq 0$ . Can someone give me any idea about how I can prove it ?I was thinking to: I know that $\mid X \mid \geq r(X)$ then I ...
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### Lattice of Flats of Graphic Matroid and Intersection Lattice of Graphic Arrangement

Let $G$ be a simple graph. We will be looking at hyperplane arrangements in $\mathbb{R}^d$. Suppose $\mathcal{H}$ is the graphic hyperplane arrangement arising from $G$. Let $L(\mathcal{H})$ be the ...
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### Definition of circuits in matroid theory

I am reading about matroids and I am kind of confused in the following definition. A nonempty subset $C$ of $\{0,1,...,n\}$ is a circuit of $I$ if $C= supp(l)$ for some nonzero linear form $l$ in ...
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### What are some interesting applications of the Bixby-Coullard inequality?

Recall that the connectivity function $\lambda_{M} = \lambda$ of a matroid $M$ may be defined as follows, letting $X$ be an arbitrary subset of the ground set of $M$ (and letting $r_{M} = r$ denote ...
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### How to find a maximum matching which considering the weight of vertex

First, if G=(V,E), and C is the collection of all vertices set which can be covered by a matching in G. I have proved that such (V,C) is a matroid. And each vertex $v_i$ has a weight $w(v_i)$, how ...
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### Matroids: Prove that for circuit $C$ and its cocircuit $C^*$: $|C \cap C^*| \neq 1$

Prove that for circuit $C$ and its cocircuit $C^*$: $|C \cap C^*| \neq 1$ Any hints and assistance would be very nice! Thank you.
### If $M$ is a matroid, and $e \in M$, than what is the closure function of $M/e$, and what is the closure function of $M\setminus e$?
If $M$ is a matroid, and $e \in M$, than what is the closure function of $M/e$, and what is the closure function of $M\setminus e$? What I find confusing to start with, is what elements does the ...