- Page 2 and 3: Author’s Declaration I hereby dec
- Page 4 and 5: Acknowledgements First and foremost
- Page 7 and 8: Table of Contents I Growth Rates 1
- Page 9 and 10: 6.4 Weighted Covers and Scatteredne
- Page 12 and 13: Chapter 1 Introduction This thesis
- Page 14 and 15: M. The nonloops of M can thus be pa
- Page 16 and 17: matroids, and L(q) for L(GF(q)) for
- Page 18 and 19: Proposition 1.1.5. If k ≥ 0 is an
- Page 20 and 21: Proof. The theorem evidently holds
- Page 22 and 23: Corollary 1.3.6. hR(n) = n+1 . 2
- Page 24 and 25: 1.5 This Thesis Part I of this thes
- Page 26 and 27: matroid PG(n − 1, q, n), obtained
- Page 28 and 29: Another way to construct classes to
- Page 30 and 31: special case of the general problem
- Page 32 and 33: Chapter 2 Projective Geometries Rec
- Page 34 and 35: If P = (X, L) is a projective space
- Page 36 and 37: Theorem 2.3.2. If M is a binary mat
- Page 38 and 39: A matching of a matroid M is a mutu
- Page 40 and 41: Proof. We show that PG + (n − 1,
- Page 42 and 43: Proof. Let P denote the set of poin
- Page 44: As M ∈ Pq,k, we may assume that N
- Page 47 and 48: Now, let M2 be a minimal contractio
- Page 49 and 50: matroids can be formed by deleting
- Page 51 and 52: Lemma 3.3.6. There is an integer-va
- Page 53 and 54:
3.3.8.1. There is a set C ′ ⊆ E

- Page 55 and 56:
Let M3 ∈ M be a matroid such that

- Page 57 and 58:
Proof of claim: Define a function g

- Page 59 and 60:
that this matroid is uniquely deter

- Page 61 and 62:
Proof. By Lemma 4.1.2, it is enough

- Page 63 and 64:
4.3.1.1. L + contains a (k + 1)-mat

- Page 65 and 66:
Lemma 4.4.1. There is an integer-va

- Page 67 and 68:
or • M ′ has a weakly round, sp

- Page 69 and 70:
• ε(M ′ ) > | PG (k) (r(M ′

- Page 71 and 72:
Proof. Set nq to be an integer larg

- Page 74 and 75:
Chapter 5 Minors and Structure Theo

- Page 76 and 77:
When M is not binary, this theorem

- Page 78 and 79:
5.4 The Growth Rate Conjecture Just

- Page 80 and 81:
Proof. The result is plain for r(M)

- Page 82 and 83:
Chapter 6 Uniform Minors I In order

- Page 84 and 85:
Lemma 6.2.1. Let d ≥ 1 be an inte

- Page 86 and 87:
There is therefore some e ∈ B suc

- Page 88 and 89:
Lemma 6.4.4. Let a ≥ 1 and d ≥

- Page 90 and 91:
Proof. Let M ′ = M| clM(S ∪ {e1

- Page 92 and 93:
as p − 1 ≥ m. Let X ′′ = F

- Page 94 and 95:
Proof of claim: By definition of pi

- Page 96 and 97:
The set Y ⊆ S ′ is simple in M

- Page 98 and 99:
Now, let N = Ns| clNs(J). We have r

- Page 100 and 101:
Lemma 6.8.2. Let a0, a1, q and d be

- Page 102 and 103:
6.8.4.1. Let 1 ≤ i ≤ j ≤ a, a

- Page 104 and 105:
Chapter 7 Uniform Minors II The goa

- Page 106 and 107:
• S| clS({e1, e ′ 1, . . . , ei

- Page 108 and 109:
(N) > 2(i − 1), and τ d (N) > µ

- Page 110 and 111:
nonloop x of Mi|(L ∩ E(R)), and f

- Page 112 and 113:
R ′ of M ′ such that B(S)∪{e,

- Page 114:
Proof. We may assume that the third

- Page 117 and 118:
minor theorem would imply that grow

- Page 119 and 120:
are the graph-like matroids arising

- Page 122 and 123:
Bibliography [1] S. Archer. Near Va

- Page 124 and 125:
[28] R. Lidl and H. Niederreiter. F

- Page 126 and 127:
Notation +F , 16 M /X, 5 M\X, 5 M/X

- Page 128:
ound, 39, 95 scattered, 78 similar,