Related papers: Strongly increasing sequences

We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete $\Pi^1_1$ set…

We will prove several expanders with exponent strictly greater than $2$. For any finite set $A \subset \mathbb R$, we prove the following six-variable expander results: \begin{align*} |(A-A)(A-A)(A-A)| &\gg…

The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is…

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…

We consider a variant of Dickson lemma, where each entry of a vector can be reseted or incremented by 1 in respect to the previous one. We give an example of non dominating sequence of length $2^{2^{\theta (n)}}$. It perfectly match the…

We show that in the theory ZF + DC + for every cardinal {\lambda}, the set of infinite subsets of {\lambda} is well-ordered (i.e., Shelah's AX4), the {\theta}-function measuring the surjective size of the powersets P({\kappa}) can take…

Let $\Omega$ be a set of positive integers and let $f:\Omega \rightarrow \Omega$ be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m \in \Omega$ has \textit{increasing-decreasing…

We give arguments for and prove the consistency of some internal forcing axioms.

The purpose of this paper is to present a general method for forcing on $\omega_2$ and $\omega_3$ with finite conditions, while preserving all cardinals and some fragments of $\mathrm{GCH}$. This method is based on the technique of forcing…

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term…

In this article we adapt the existing account of class-forcing over a ZFC model to a model $(M,\mathcal{C})$ of Morse-Kelley class theory. We give a rigorous definition of class-forcing in such a model and show that the Definability Lemma…

Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real $\eta \in \prod_{i<\omega} n^*_i$, which is eventually different from any old real in $\prod_{i<\omega} n^*_i$, where the…

A strongly polynomial sequence of graphs $(G_n)$ is a sequence $(G_n)_{n\in\mathbb{N}}$ of finite graphs such that, for every graph $F$, the number of homomorphisms from $F$ to $G_n$ is a fixed polynomial function of $n$ (depending on $F$).…

Let F be a fixed finite field of characteristic at least 5. Let G = F^n be the n-dimensional vector space over F, and write N := |G|. We show that if A is a subset of G with size at least c_F N(log N)^{-c}, for some absolute constant c > 0…

Let \alpha be a countable ordinal and \P(\alpha) the collection of its subsets isomorphic to \alpha. We show that the separative quotient of the set \P (\alpha) ordered by the inclusion is isomorphic to a forcing product of iterated reduced…

We present a detailed general framework to describe the forcing $\tilde{\mathbb{E}}$, defined by Kellner, Shelah and Tan\u{a}sie to prove the consistency with ZFC of an alternative order of Cicho\'n's maximum. Our presentation is close to…

Let $F$ be a distribution function on the line in the domain of attraction of a stable law with exponent $\alpha\in(0,1/2]$. We establish the strong renewal theorem for a random walk $S_1,S_2,\ldots$ with step distribution $F$, by extending…

We examine the Zermelo Fraenkel set theory with Choice (ZFC) enhanced by one of the (structural) reflection principles down to a small cardinal and/or Recurrence Axioms defined below. The strongest forms of reflection principles spotlight…

We introduce a simple but powerful technique to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to non conservative zero range processes on finite graphs, and to…

We show that the forcing axiom for countably compact, $\omega_2$-Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah and sets a new limit to the generalization of Martin's Axiom to the stage…