Related papers: Combinatorial properties of Hechler forcing
This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 :…
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…
Any model of ZFC + GCH has a generic extension (made with a poset of size aleph_2) in which the following hold: MA + 2^{aleph_0}= aleph_2+ there exists a Delta^2_1-well ordering of the reals. The proof consists in iterating posets designed…
Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main…
We show that it is consistent relative to ZF, that there is no well-ordering of $\mathbb{R}$ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we…
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove…
We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers $k\geq1$, $r\geq 0$, and $\ell\geq (r+1)r$, and for any…
We show the consistency of ZFC +''there is no NWD-ultrafilter on omega'', which means: for every non principle ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for…
A conjecture by the second author, proven by Bonnaf\'e-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the…
We prove a conjecture of Comon and Ottaviani that typical real Waring ranks of bivariate forms of degree $d$ take all integer values between $\lfloor \frac{d+2}{2}\rfloor$ and $d$. That is we show that for all $d$ and all $\lfloor…
The Lefschetz question asks if multiplication by a power of a general linear form, $L$, on a graded algebra has maximal rank (in every degree). We consider a quotient by an ideal that is generated by powers of linear forms. Then the…
We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that…
A group has normal rank (or weight) greater than one if no single element normally generates the group. The Wiegold problem from 1976 asks about the existence of a finitely generated perfect group of normal rank greater than one. We show…
Cicho\'n's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's, Goldstern, Kellner and Shelah constructed a forcing…
A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…
This paper presents a first result of a long term research project dealing with the construction of d-orthogonal polynomials with Hahn's property. We shall show that the latter class could be characterized by expanding a polynomial as a…
A seminal result by Koml\'os, Sark\"ozy, and Szemer\'edi states that if a graph $G$ with $n$ vertices has minimum degree at least $kn/(k + 1)$, for some $k \in \mathbb{N}$ and $n$ sufficiently large, then it contains the $k$-th power of a…
Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that…
The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation…
A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of…