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## Matroid theory and Hrushovski’s predimension construction (2011)

Citations: | 1 - 0 self |

### Citations

704 |
Matroid theory
- Oxley
- 1992
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Citation Context ...struction from [8]. All of this is well-known: the proofs which we have omitted can be found in [8] or [15] (though most of them are oneline proofs which are easy to reconstruct). We use Oxley’s book =-=[14]-=- as our basic reference on matroid theory. The paper [9] by Ingleton is a very clear survey of results on strict gammoids. Suppose A is a set and R is a set of finite, non-empty subsets of A (in fact ... |

47 |
Transversal Theory
- Mirsky
- 1971
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Citation Context ... of closed sets in (A, cl). We first show that there is an α-transversal of F . By a generalization of Hall’s Marriage Theorem (quoted in [12] as due to Welsh, but attributed to Halmos and Vaughan in =-=[13]-=-), it will suffice to prove that for distinct F1, . . . , Fr ∈ F we have | ⋃ i≤r Fi| ≥ ∑ i≤r α(Fi). If the union is not one of the Fi, then | ⋃ i Fi| = α( ⋃ i Fi) + d( ⋃ i Fi) + ∑ F< ⋃ i Fi α(F ) ≥ ∑ ... |

21 |
A new strongly minimal set, Annals of Pure and Applied Logic 62
- Hrushovski
- 1993
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Citation Context ... a finite pregeometry. VERSION: 17 MAY 2011 1 ar X iv :1 10 5. 38 22 v1s[ ma th. LO ]s19sM ays20 11 2 DAVID M. EVANS We begin with an exposition of the basic Hrushovski predimension construction from =-=[8]-=-. All of this is well-known: the proofs which we have omitted can be found in [8] or [15] (though most of them are oneline proofs which are easy to reconstruct). We use Oxley’s book [14] as our basic ... |

16 |
Gammoids and transversal matroids
- Ingleton, Piff
- 1973
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Citation Context ...t the matroids of the form PG(A) for A ∈ C are exactly the duals of the transversal matroids (see 1.6 of [14]). We give a short proof of this result in this section. By a theorem of Ingleton and Piff =-=[10]-=-, the duals of transversal matroids are the strict gammoids defined by Mason in [12]. So the class of matroids PG(C) = {PG(A) : A ∈ C} appears in the literature as the class of strict gammoids, or cot... |

13 |
On some combinatorial properties of algebraic matroids
- Dress, Lovász
- 1987
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Citation Context ...d ā is a tuple of elements of A, then there is a closed B0 ⊆ B such that for every closed B1 ⊆ B we have d(ā/B1) = d(ā/B) iff B0 ⊆ B1. Such a property was considered for full algebraic matroids in =-=[2]-=-. We shall show: Theorem 6.1. If A is a strict gammoid, then A has weak canonical bases over closed sets. For structures given by Hrushovski’s construction, this sort of result is essentially folklore... |

9 | Trivial stable structures with non-trivial reducts
- Evans
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Citation Context ...ture as the class of strict gammoids, or cotransversal matroids. Some connection between transversals and the Hrushovski constructions had already been noted and used in model theory: for example, in =-=[1, 3]-=-. Suppose A is a set and R is a set of finite, non-empty subsets of A. A transveral of (A;R) is an injective function t : R→ A with t(r) ∈ r 4 DAVID M. EVANS for all r ∈ R. Abusing terminology, we sha... |

6 | Transversal matroids and related structures - Ingleton - 1976 |

4 | Geometries of Hrushovski Constructions
- Ferreira
- 2009
(Show Context)
Citation Context ...y of nonisomorphic countable dimensional geometries; (B) one of the few properties we know about the geometries, namely flatness, does not look to be very convenient to use. However, [4] and [5] (and =-=[6]-=-) show that there are only countably many local isomorphism types of countable dimensional geometries (of strongly minimal sets) produced in [8]: by this we mean the isomorphism type of the geometry o... |

4 |
On a class of matroids arising from paths
- Mason
- 1972
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Citation Context ...matroids (see 1.6 of [14]). We give a short proof of this result in this section. By a theorem of Ingleton and Piff [10], the duals of transversal matroids are the strict gammoids defined by Mason in =-=[12]-=-. So the class of matroids PG(C) = {PG(A) : A ∈ C} appears in the literature as the class of strict gammoids, or cotransversal matroids. Some connection between transversals and the Hrushovski constru... |

3 |
Ehud Hrushovski, Pascal Koiran and Bruno Poizat, La limite des théories de courbes génériques
- Chapuis
(Show Context)
Citation Context ...ture as the class of strict gammoids, or cotransversal matroids. Some connection between transversals and the Hrushovski constructions had already been noted and used in model theory: for example, in =-=[1, 3]-=-. Suppose A is a set and R is a set of finite, non-empty subsets of A. A transveral of (A;R) is an injective function t : R→ A with t(r) ∈ r 4 DAVID M. EVANS for all r ∈ R. Abusing terminology, we sha... |

3 | The geometry of Hrushovski constructions, I. The uncollapsed case. Annals of Pure and Applied Logic 162
- Evans, Ferreira
- 2011
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Citation Context ...ther hand, (A;R) is a presentation of A so d(C) ≤ δ(C) = |C| − |(R|C)|. Thus |(R|C)| ≤ |RC | and the equality follows. This enables us to give a different proof of the finite case of Theorem 4.3 of =-=[4]-=-. Corollary 5.3. Suppose m < n and (A;R) ∈ C is such that there exists C ≤ A with |C| = n, d(C) = n − 1 and every (n − 1)-subset of C is independent. If (B;R′) ∈ Cm, then PG(A;R) is not isomorphic to ... |

3 | The geometry of Hrushovski constructions II. The strongly minimal case. Submitted
- Evans, Ferreira
- 2010
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Citation Context ...ght optimistically conjecture that H = Pk. As a first step towards this question, one might consider those H which are closed under free amalgamation. We review briefly some of the results in [4] and =-=[5]-=- using this terminology. Let Gk be the generic structure for Pk. Theorems 6.9 and 4.3 of [4] show that these pregeometries are non-isomorphic for different values of k. Theorem 5.5 of [4] shows that G... |

3 |
A survey of Jordan groups
- Macpherson
- 1994
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Citation Context ... b2 (and which preserves k with respect to the Pi). Finally we claim that there is an automorphism of P which fixes all of cl(A) and takes b1 to b2. An argument similar to that given on pp. 96–97 of =-=[11]-=- can be used. So the following is relevant to Question 7.1: Question 7.3. What are the subclassesH of Pk = (PG(Ck),k) which are amalgamation classes? Here we might want to assume that there is no b... |

2 |
convexity and notions of freeness in combinatorial geometries, Algebra Universalis
- Holland, Flatness
- 1999
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Citation Context ...ted’ from the F̃i (over ⋃ i Fi). Further results on freeness in flat pregeometries (which do not depend on their characterization in terms of Hrushovski constructions) can be found in Holland’s paper =-=[7]-=- Suppose (B1;R1), (B2;R2) ∈ C (or even in C̄) and A ≤ (Bi;Ri) (with R1|A = R2|A). We can assume A = B1 ∩ B2 and let C = B1 ∪ B2. Consider C as a set system with relations R = R1 ∪ R2. Then it can be s... |

2 | Relational structures and dimensions. In : R. Kaye et D. Macpherson (eds), Automorphism Groups of First-order Structures - Wagner - 1994 |