A Generalization of the Erdős-Turán Law . . .

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős-Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths

Neural function approximation on graphs: shape modelling, graph discrimination & compression

Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces

Complete type amalgamation and Roth's theorem on arithmetic progressions

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers and of Nikolov-Pyber, on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. Furthermore, we bound the number of solutions to certain equations, such as $x^n\cdot y^m=z^k$ for $n+m=k$, in subsets of small tripling in groups. In particular, we show the existence of lower bounds on the number of arithmetic progressions of length $3$ for subsets of small doubling without involutions in arbitrary abelian groups

Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings

This thesis consists of three independent parts. The first part of the thesis is concerned with Ramsey theory. Given an integer $q\geq 2$, a graph $G$ is said to be \emph{$q$-Ramsey} for another graph $H$ if in any $q$-edge-coloring of $G$ there exists a monochromatic copy of $H$. The central line of research in this area investigates the smallest number of vertices in a $q$-Ramsey graph for a given $H$. In this thesis, we explore two different directions. First, we will be interested in the smallest possible minimum degree of a minimal (with respect to subgraph inclusion) $q$-Ramsey graph for a given $H$. This line of research was initiated by Burr, Erdős, and Lovász in the 1970s. We study the minimum degree of a minimal Ramsey graph for a random graph and investigate how many vertices of small degree a minimal Ramsey graph for a given $H$ can contain. We also consider the minimum degree problem in a more general asymmetric setting. Second, it is interesting to ask how small modifications to the graph $H$ affect the corresponding collection of $q$-Ramsey graphs. Building upon the work of Fox, Grinshpun, Liebenau, Person, and Szabó and Rödl and Siggers, we prove that adding even a single pendent edge to the complete graph $K_t$ changes the collection of 2-Ramsey graphs significantly. The second part of the thesis deals with orthogonal Latin squares. A {\em Latin square of order $n$} is an $n\times n$ array with entries in $[n]$ such that each integer appears exactly once in every row and every column. Two Latin squares $L$ and $L'$ are said to be {\em orthogonal} if, for all $x,y\in [n]$, there is a unique pair $(i,j)\in [n]^2$ such that $L(i,j) = x$ and $L'(i,j) = y$; a system of {\em $k$ mutually orthogonal Latin squares}, or a {\em $k$-MOLS}, is a set of $k$ pairwise orthogonal Latin squares. Motivated by a well-known result determining the number of different Latin squares of order $n$ log-asymptotically, we study the number of $k$-MOLS of order $n$. Earlier results on this problem were obtained by Donovan and Grannell and Keevash and Luria. We establish new upper bounds for a wide range of values of $k = k(n)$. We also prove a new, log-asymptotically tight, bound on the maximum number of other squares a single Latin square can be orthogonal to. The third part of the thesis is concerned with grid coverings with multiplicities. In particular, we study the minimum number of hyperplanes necessary to cover all points but one of a given finite grid at least $k$ times, while covering the remaining point fewer times. We study this problem for the grid $\mathbb{F}_2^n$, determining the number exactly when one of the parameters $n$ and $k$ is much larger than the other and asymptotically in all other cases. This generalizes a classic result of Jamison for $k=1$. Additionally, motivated by the recent work of Clifton and Huang and Sauermann and Wigderson for the hypercube $\set{0,1}^n\subseteq\mathbb{R}^n$, we study hyperplane coverings for different grids over $\mathbb{R}$, under the stricter condition that the remaining point is omitted completely. We focus on two-dimensional real grids, showing a variety of results and demonstrating that already this setting offers a range of possible behaviors.Diese Dissertation besteht aus drei unabh\"angigen Teilen. Der erste Teil beschäftigt sich mit Ramseytheorie. Für eine ganze Zahl $q\geq 2$ nennt man einen Graphen \emph{$q$-Ramsey} f\"ur einen anderen Graphen $H$, wenn jede Kantenf\"arbung mit $q$ Farben einen einfarbigen Teilgraphen enthält, der isomorph zu $H$ ist. Das zentrale Problem in diesem Gebiet ist die minimale Anzahl von Knoten in einem solchen Graphen zu bestimmen. In dieser Dissertation betrachten wir zwei verschiedene Varianten. Als erstes, beschäftigen wir uns mit dem kleinstm\"oglichen Minimalgrad eines minimalen (bezüglich Teilgraphen) $q$-Ramsey-Graphen f\"ur einen gegebenen Graphen $H$. Diese Frage wurde zuerst von Burr, Erd\H{o}s und Lov\'asz in den 1970er-Jahren studiert. Wir betrachten dieses Problem f\"ur einen Zufallsgraphen und untersuchen, wie viele Knoten kleinen Grades ein Ramsey-Graph f\"ur gegebenes $H$ enthalten kann. Wir untersuchen auch eine asymmetrische Verallgemeinerung des Minimalgradproblems. Als zweites betrachten wir die Frage, wie sich die Menge aller $q$-Ramsey-Graphen f\"ur $H$ verändert, wenn wir den Graphen $H$ modifizieren. Aufbauend auf den Arbeiten von Fox, Grinshpun, Liebenau, Person und Szabó und Rödl und Siggers beweisen wir, dass bereits der Graph, der aus $K_t$ mit einer h\"angenden Kante besteht, eine sehr unterschiedliche Menge von 2-Ramsey-Graphen besitzt im Vergleich zu $K_t$. Im zweiten Teil geht es um orthogonale lateinische Quadrate. Ein \emph{lateinisches Quadrat der Ordnung $n$} ist eine $n\times n$-Matrix, gef\"ullt mit den Zahlen aus $[n]$, in der jede Zahl genau einmal pro Zeile und einmal pro Spalte auftritt. Zwei lateinische Quadrate sind \emph{orthogonal} zueinander, wenn f\"ur alle $x,y\in[n]$ genau ein Paar $(i,j)\in [n]^2$ existiert, sodass es $L(i,j) = x$ und $L'(i,j) = y$ gilt. Ein \emph{k-MOLS der Ordnung $n$} ist eine Menge von $k$ lateinischen Quadraten, die paarweise orthogonal sind. Motiviert von einem bekannten Resultat, welches die Anzahl von lateinischen Quadraten der Ordnung $n$ log-asymptotisch bestimmt, untersuchen wir die Frage, wie viele $k$-MOLS der Ordnung $n$ es gibt. Dies wurde bereits von Donovan und Grannell und Keevash und Luria studiert. Wir verbessern die beste obere Schranke f\"ur einen breiten Bereich von Parametern $k=k(n)$. Zusätzlich bestimmen wir log-asymptotisch zu wie viele anderen lateinischen Quadraten ein lateinisches Quadrat orthogonal sein kann. Im dritten Teil studieren wir, wie viele Hyperebenen notwendig sind, um die Punkte eines endlichen Gitters zu überdecken, sodass ein bestimmter Punkt maximal $(k-1)$-mal bedeckt ist und alle andere mindestens $k$-mal. Wir untersuchen diese Anzahl f\"ur das Gitter $\mathbb{F}_2^n$ asymptotisch und sogar genau, wenn eins von $n$ und $k$ viel größer als das andere ist. Dies verallgemeinert ein Ergebnis von Jamison für den Fall $k=1$. Au{\ss}erdem betrachten wir dieses Problem f\"ur Gitter im reellen Vektorraum, wenn der spezielle Punkt überhaupt nicht bedeckt ist. Dies ist durch die Arbeiten von Clifton und Huang und Sauermann und Wigderson motiviert, die den Hyperwürfel $\set{0,1}^n\subseteq \mathbb{R}^n$ untersucht haben. Wir konzentrieren uns auf zwei-dimensionale Gitter und zeigen, dass schon diese sich sehr unterschiedlich verhalten können

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Spatial discretizations of generic dynamical systems

How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations? To try to answer this question, we study in this manuscript a model reflecting what happens when the orbits of a discrete time system $f$ (for example an homeomorphism) are computed numerically . The computer working in finite numerical precision, it will replace $f$ by a spacial discretization of $f$, denoted by $f_N$ (where the order $N$ of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps $f_N$ for a generic system $f$ and $N$ going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or $C^1$-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations $f_N$, when $f$ is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of $f$ from that of a single discretization $f_N$ : its dynamics strongly depends on the order $N$. To detect some dynamical features of $f$, we have to consider all the discretizations $f_N$ when $N$ goes through $\mathbf N$. The second part deals with the linear case, which plays an important role in the study of $C^1$-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part, though weaker and harder to prove.Comment: 322 pages. This is an improved version of the thesis of the author (among others, the introduction and conclusion have been translated into English). In particular, it contains works already published on arXiv. Comments welcome