Related papers: Monadic second-order properties of very sparse ran…

It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number $(G) of nested quantifiers in a such formula can serve as a measure for the ``first order…

The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…

We study on which classes of graphs first-order logic (FO) and monadic second-order logic (MSO) have the same expressive power. We show that for all classes C of graphs that are closed under taking subgraphs, FO and MSO have the same…

We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed…

When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.…

We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma$ and a first order theory $T$ for $\sigma$ composed of symmetry and…

For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO…

In the paper, we prove that existential monadic second order convergence law fails for the binomial random graph $G(n,n^{-\alpha})$ for every $\alpha\in(0,1)$.

Let $\alpha\in(0,1)_\mathbb{R}$ be irrational and $G_n = G_{{n, 1/n}^\alpha}$ be the random graph with edge probability $1/n^\alpha$; we know that it satisfies the 0-1 law for first order logic. We deal with the failure of the 0-1 law for…

We analyze some local properties of sparse Erdos-Renyi graphs, where $d(n)/n$ is the edge probability. In particular we study the behavior of very short paths. For $d(n)=n^{o(1)}$ we show that $G(n,d(n)/n)$ has asymptotically almost surely…

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped…

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…

Let G_n be the random graph on [n]= {1, ...,n} with the possible edge {i,j} having probability being p_{|i-j|}= 1/|i-j|^alpha, alpha in (0,1) irrational. We prove that the zero one law (for first order logic) holds. The paper is continued…

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…

For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$…

Let $d,n\in \mathbb{N}$ be such that $d=\omega(1)$, and $d\le n^{1-a}$ for some constant $a>0$. Consider a $d$-regular graph $G=(V, E)$ and the random graph process that starts with the empty graph $G(0)$ and at each step $G(i)$ is obtained…

Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory,…

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of…

We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $n+1$, the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with…