Global ETD Search
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Showing 1 to 20 of 29 for “"extremal combinatorics"”.
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Problems in extremal combinatorics
We consider a variety of problems in extremal graph and set theory. Given a property $\Gamma$ and a family of sets ${\mathcal F}$, let $f({\mathcal F},\Gamma)$ be the size of the largest subfamily of ${\mathcal F}$ having property $\Gamma$. Let $f(m,\Gamma)$ be the minimum of $f({\mathcal …
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Extremal Combinatorics and Universal Algorithms
… different areas of mathematics: automata theory, combinatorics of partially ordered sets and extremal combinatorics. Firstly, we focus on some new automata that do not seem to have occurred much in the literature, that of solvability of mazes. For our model, a maze is a countable strongly …
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Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive combinatorics
… we answer a question of Nathanson in additive combinatorics about sums, differences and products of sets in $\mathbb{Z}_N$ (the integers modulo $N$). For all $\epsilon>0$ and $k\in\mathbb{N}$, we construct a subset $A\subset\mathbb{Z}_N$ for some $N$, such that $|A^2+kA|\leq\epsilon N$, while …
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Inducibility and Subgraph Density Problems in Graphs
In this thesis, we consider problems in extremal combinatorics concerning the number of copies of a fixed graph in another larger graph. Many of these problems can be phrased in terms of inducibility, i.e. the maximum proportion of induced copies of the smaller graph F in a larger graph G. We also …
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Random and exact structures in combinatorics
… to notions of randomness and structure in combinatorics and probability. One central notion, the pseudorandomness-structure dichotomy, has played a key role in additive combinatorics and extremal graph theory. More generally, however, such notions come into play in the study of …
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Forbidden substructures: induced subgraphs, Ramsey games, and sparse hypergraphs
We study problems in extremal combinatorics with respect to forbidden induced subgraphs, forbidden colored subgraphs, and forbidden subgraphs. In Chapter 2, we determine exactly which graphs H have the property that almost every H-free graph has a vertex partition into k cliques and independent …
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Testability of linear-invariant properties
… theory, computational learning theory, and extremal combinatorics. In the history of the area, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. …
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Sparse regularity and relative Szemerédi theorems
… regularity lemma, a fundamental tool in extremal combinatorics. The regularity method, in its original form, is effective only for dense graphs. It has been a long standing problem to extend the regularity method to sparse graphs. We solve this problem by proving a so-called "counting …
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Seedless Extractors
… to cryptography, complexity theory, and combinatorics. As it is impossible to construct a single extractor that works for all weak sources of randomness, research on extractors has split into two complementary settings: (1) the seeded setting, where the extractor is equipped with a short …
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Extremal graph theory: supersaturation and enumeration
… supersaturation and enumeration problems in extremal combinatorics. In Chapter 2, with Balogh, we disprove a conjecture of Erdos and Tuza concerning the number of different ways one can create a copy of K_4, a complete graph on 4 vertices, in a K_4-free graph. In Chapter 3, we extend a …
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Enumerating combinatorial objects with limited sub-configurations
Many well-studied problems in extremal combinatorics concern the number and the typical structure of discrete objects with forbidden substructures. Over the past decades, such problems have been extensively studied for various objects by many notable researchers. This thesis focuses on several …
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Combinatorial Problems on the Integers: Colorings, Games, and Permutations
… integers. These problems fit inside the areas of extremal combinatorics and enumerative combinatorics.</p> <p>We first study monochromatic solutions to equations when integers are colored with finitely many colors in Chapter 2. By looking at subsets of {1, 2, . . . , <em>n</em>} whose least common …
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Extremal problems on counting combinatorial structures
The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on $n$ …
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Algorithms and Algorithmic Barriers in High-Dimensional Statistics and Random Combinatorial Structures
… algorithms are based on Ramsey Theory from extremal combinatorics. To the best of our knowledge, this is the first usage of Ramsey Theory to show algorithmic hardness for models with random parameters. 2. Our second focus is on the Sherrington-Kirkpatrick (SK) spin glass model, a mean-field …
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Explicit pseudorandom distributions for restricted models of computation
Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-02-04 without embargo terms
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On some problems in extremal, probabilistic and enumerative combinatorics
… selection of problems from various areas of Combinatorics and Graph Theory, a fast developing field that provides a diverse spectrum of powerful tools with numerous applications to computer science, optimization theory and economics. In this thesis, we focus on extremal, probabilistic and …
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Extremal problems in the cube and the grid and other combinatorial results
… contains results from various areas of combinatorics. In Chapters 2, 3 and 4 we consider questions in the area of isoperimetric inequalities. In Chapter 2, we find the exact classification of all subsets A⊆{0,1}^n for which both A and A^c minimise the size of the neighbourhood, which …
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Poset saturation and other combinatorial results
In this dissertation we discuss a number of combinatorial results. These results fall into four broad areas: poset saturation, Ramsey theory, pursuit and evasion, and union-closed families. Chapter 2 is dedicated to the area of poset saturation. Given a finite poset P, we call a family F of subsets …
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