相关论文: Spaces of completions of elementary theories and c…

This is an extended version of the thesis presented to the Programa de P\'os-Gradua\c{c}\~ao em Matem\'atica of the Departamento de Matem\'atica, PUC-Rio, in September 2013, incorporating some suggestions from the examining commission.…

In this work limit probabilities of first-order properties of the random $s$-uniform hypergraph in the binomial model $G^{s}(n,p)$ are studied. We give a complete discription of all positive $\alpha$ such that $G^{s}(n,n^{-\alpha})$ obeys…

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a…

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds.…

For a sequence p=(p(1),p(2), ...) let G(n,p) denote the random graph with vertex set {1,2, ...,n} in which two vertices i, j are adjacent with probability p(|i-j|), independently for each pair. We study how the convergence of probabilities…

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…

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 establish central and local limit theorems for the number of vertices in the largest component of a random $d$-uniform hypergraph $\hnp$ with edge probability $p=c/\binnd$, where $(d-1)^{-1}+\eps<c<\infty$. The proof relies on a new,…

Fix a sequence of $d$-regular graphs $(G_d)_{d\in \mathbb{N}}$ and denote by $G_{d,p}$ the graph obtained from $G_d$ after edge-percolation with probability $p=c/d$, for a constant $c>0$. We prove a quantitative local convergence of…

In the classical Erd\"os-R\'enyi random graph G(n,p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n,p) is homogeneous in the sense that all vertices have the same…

Let P_{n,d,D} denote the graph taken uniformly at random from the set of all labelled planar graphs on {1,2,...,n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability…

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$,…

We consider a number $\nu_n$ of components in a random graph $G(n,p)$ with $n$ vertices, where the probability of an edge is equal to $p$. By operating with special generating functions we shows the next asymptotic relation for factorial…

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from…

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have…

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…

Fix a positive integer $n$, a real number $p\in (0,1]$, and a (perhaps random) hypergraph $\mathcal{H}$ on $[n]$. We introduce and investigate the following random multigraph model, which we denote $\mathbb{G}(n,p\, ; \,\mathcal{H})$: begin…

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial…