Thesis

### Problems in extremal and probabilistic combinatorics: cubes, squares and permutations

Abstract:

We begin by studying the possible intersection sizes of a $k$-dimensional linear subspace with the hypercube $\{0,1\}^n$. For a fixed $k$, the largest intersection size is $2^k$ and it was shown by Melo and Winter that the second largest intersection size is $2^{k-1} + 2^{k-2}$. We show that all intersections sizes larger than $2^{k-1}$ are of the form $2^{k-1} + 2^{i}$ or $35 \cdot 2^{k-6}$, completely determining the large'' intersection sizes and disproving a conjecture of Melo and Wi...

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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Oxford college:
Role:
Author
ORCID:
0000-0002-4119-4599

#### Contributors

Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Oxford college:
Merton College
Role:
Supervisor
Institution:
G-SCOP Laboratory
Oxford college:
Merton College
Role:
Examiner
Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Oxford college:
Merton College
Role:
Examiner
Type of award:
DPhil
Level of award:
Doctoral
Awarding institution:
University of Oxford
Language:
English
Keywords:
Subjects:
Deposit date:
2022-02-15