Results in Extremal Graph Theory, Ramsey Theory and Additive Combinatorics

This dissertation contains results from various areas of Combinatorics. In Chapter 2, we consider a central problem in Extremal Graph Theory. The extremal number (or Turán number) ex(n,H) of a graph H is the maximum number of edges in an H-free graph on n vertices. It is a major area of research to better understand the extremal number of bipartite graphs. In this chapter we develop a new method which allows us to obtain strong (and often best possible) upper bounds in a wide range of cases. Our results answer several conjectures of Conlon and Lee, and Kang, Kim and Liu. Furthermore, they relate to and improve work of (among others) Füredi, Alon, Krivelevich and Sudakov, Kostochka and Pyber, and Jiang and Seiver. While in Chapter 2 the focus is on subdivided graphs, in Chapter 3 we study the extremal number of blow-ups. In particular, we obtain tight upper bounds for the extremal number of blow-ups of trees. As an extension of this, we pose a general conjecture relating the extremal number of F and that of its blow-up. We prove the conjecture for the 2-blowup of C_6. In Chapter 4 we study a coloured variant of the Turán problem. The rainbow Turán number of H, denoted by ex*(n,H), is the maximum possible number of edges in an n-vertex properly edge-coloured graph without a rainbow subgraph isomorphic to H. We prove that ex*(n,C_{2k})=O(n^{1+1/k}), which is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraete. We use the same method to answer several further questions in various topics: among others, a question of Conlon and Tyomkyn on colour-isomorphic cycles and a conjecture of Jiang and Newman of blow-ups of cycles. We also disprove an old conjecture of Erdős and Simonovits on (ordinary) extremal numbers. In Chapter 5, we consider the following problem. Let 2 ≤ s < t be fixed integers. If G is an arbitrary K_t-free graph on n vertices, how large a K_s-free induced subgraph must there exist in G? This number, which is a generalisation of the usual off-diagonal Ramsey numbers, is viewed as a function in n, and is called the Erdős-Rogers function. We obtain new upper bounds in the case s+2 ≤ t ≤ 2s-1, improving results of (among others) Bollobás, Erdős and Krivelevich, and answering a question of Dudek, Retter and Rödl. In Chapter 6, we investigate the relationship between two well-studied notions of tensor rank. We show that the partition rank of a tensor is bounded above by a polynomial in the analytic rank of the same tensor. This improves Ackermann-type bounds obtained by various authors including Green and Tao, and Bhowmick and Lovett. In Chapter 7, we use the main technical lemma of Chapter 6 to prove a result about the expansion of subsets of the Cayley graph on the tensor product F_2^{n_1} ⊗ … ⊗ F_2^{n_d} where the generators are the rank 1 tensors. This is motivated by the famous Unique Games Conjecture from Theoretical Computer Science, and is a partial generalisation of a recent breakthrough result of Khot, Minzer and Safra. In Chapter 8, we ask the following question. Given constants α,β,γ, what is the minimal possible edge density of a graph G on n vertices with the property that every subset A⊆V(G) with |A| ≥ αn contains a subset B⊆A with |B| ≥ βn such that G[B] has edge density at least γ? We also study a bipartite version of this question, obtaining sharp results in both cases. In Chapter 9, we determine asymptotically the maximum possible number of induced C_5's in a planar graph on n vertices

Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity

Unexpected Stein fillings, rational surface singularities, and plane curve arrangements

We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong-van Straten in [dJvS98]; they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong-van Straten's construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most -5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).Comment: 70 pages, 29 figures, additions to v2 are Theorem 1.3 and its corresponding discussion in 4.3, along with added references, v3 updated Remark 4.

Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity

Nonarchimedean Holographic Entropy from Networks of Perfect Tensors

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua