Related papers: Rothberger's property in finite powers
In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Ne\v{s}et\v{r}il and V. R\"{o}dl claims that the class of all finite posets…
In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all…
It has become obvious in the recent development that the structural Ramsey property is a categorical property: it depends not only on the choice of objects, but also on the choice of morphisms involved. In this paper we explicitely put the…
We show the finiteness of perfect powers in orbits of polynomial dynamical systems over an algebraic number field. We also obtain similar results for perfect powers represented by ratios of consecutive elements in orbits. Assuming the…
We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of…
We introduce the forcing property "almost strong properness" which sits between properness and strong properness. As an application, we introduce a simple forcing with finite conditions to force $\rm MRP$.
In 2012 M. Soki\'c proved that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. B\"ottcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another…
We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the…
We investigate the behavior of countable Borel equivalence relations (CBERs) on topological Ramsey spaces. First, we give a simple proof of the fact that every CBER on $[\mathbb{N}]^{\mathbb{N}}$ is hyperfinite on some set of the form…
We consider finitary approximations of the (embedding) Ramsey property. Using a class of homogeneous reducts of random ordered hypergraphs, we prove that these properties form a strict hierarchy. We also show that every class of finite…
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete…
In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result…
Every statement of the Ramsey theory of finite structures corresponds to the fact that a particular category has the Ramsey property. We can, then, compare the strength of Ramsey statements by comparing the ``Ramsey strength'' of the…
We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Soki\'c. As a bonus, our proof…
In the paper we prove for every finite algebra A that either it has the polynomially generated powers (PGP) property, or it has the exponentially generated powers (EGP) property. For idempotent algebras we give a simple criteria for the…
We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results…
In this paper we investigate the range of validity of Ruelle's property. First, we show that every finitely-generated Fuchsian group has Ruelle's property. We also prove the existence of an infinitely-generated Fuchsian group satisfying…