Related papers: Random graphs in the monadic theory of order
We prove that both multiple Rademacher system and Rademacher chaos possess the property of random unconditional convergence in the space $L_\infty$. This fact combined with some intimate connections between $L_\infty$-norms of linear…
Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO is fixed-parameter…
The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher order…
In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and J.A.…
We prove that for any monotone class of finite relational structures, the first-order theory of the class is NIP in the sense of stability theory if, and only if, the collection of Gaifman graphs of structures in this class is nowhere…
This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of…
Reiher, R\"odl, Sales, and Schacht initiated the study of relative Tur\'an densities of ordered graphs and showed that it is more subtle and interesting than the unordered case. For an ordered graph $F$, its relative Tur\'an density,…
Higher order set theory has been a topic of interest for some time, with recent efforts focused on the strength of second order set theories [KW16]. In this paper we strive to present one 'theory of collections' that allows for a formal…
In this paper, we study arbitrary models of the first-order theory of a ring $A$ where the additive group $A$ is a finitely generated abelian group. Following an earlier paper by this author, Alexei G. Myasnikov and Francis Oger, we call…
We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group…
A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Given any finite graph $H$, we prove that every one-ended, unimodular random rooted graph that does not have H as a minor must be…
The theory of random real numbers is exceedingly well-developed, and fascinating from many points of view. It is also quite challenging mathematically. The present notes are intended as no more than a gateway to the larger theory. They…
Courcelle's Theorem states that on graphs $G$ of tree-width at most $k$ with a given tree-decomposition of size $t(G)$, graph properties $\mathcal{P}$ definable in Monadic Second Order Logic can be checked in linear time in the size of…
Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The…
An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the…
There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a…
We study the problem of distinguishing between two independent samples $\mathbf{G}_n^1,\mathbf{G}_n^2$ of a binomial random graph $G(n,p)$ by first order (FO) sentences. Shelah and Spencer proved that, for a constant $\alpha\in(0,1)$,…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…
We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph…
For many standard models of random structure, first-order logic sentences exhibit a convergence phenomenon on random inputs. The most well-known example is for random graphs with constant edge probability, where the probabilities of…