arXiv subjects

Konstantinos Tyros

Publications and source records attributed to Konstantinos Tyros.

29 records · Page 2Linked to original sources

arXiv2014

Measurable events indexed by words

For every integer $k\geq 2$ let $[k]^{<\mathbb{N}}$ be the set of all words over $k$, that is, all finite sequences having values in $[k]:=\{1,...,k\}$. A Carlson-Simpson tree of $[k]^{<\mathbb{N}}$ of dimension $m\geq 1$ is a subset of $[k]^{<\mathbb{N}}$ of the form \[ \{w\}\cup \big\{w^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_{n}(a_n): n\in \{0,...,m-1\} \text{ and } a_0,...,a_n\in [k]\big\} \] where $w$ is a word over $k$ and $(w_n)_{n=0}^{m-1}$ is a finite sequence of left variable words over $k$. We study the behavior of a family of measurable events in a probability space indexed by the elements of a Carlson-Simpson tree of sufficiently large dimension. Specifically we show the following. For every integer $k\geq 2$, every $0<\varepsilon\leq 1$ and every integer $n\geq 1$ there exists a strictly positive constant $θ(k,\varepsilon,n)$ with the following property. If $m$ is a given positive integer, then there exists an integer $\mathrm{Cor}(k,\varepsilon,m)$ such that for every Carlson--Simpson tree $T$ of $[k]^{<\mathbb{N}}$ of dimension at least $\mathrm{Cor}(k,\varepsilon,m)$ and every family $\{A_t:t\in T\}$ of measurable events in a probability space $(Ω,Σ,μ)$ satisfying $μ(A_t)\geq \varepsilon$ for every $t\in T$, there exists a Carlson--Simpson tree $S$ of dimension $m$ with $S\subseteq T$ and such that for every nonempty $F\subseteq S$ we have \[μ\Big(\bigcap_{t\in F} A_t\Big) \geq θ(k,\varepsilon,|F|). \] The proof is based, among others, on the density version of the Carlson--Simpson Theorem established recently by the authors, as well as, on a partition result -- of independent interest -- closely related to the work of T. J. Carlson, and H. Furstenberg and Y. Katznelson. The argument is effective and yields explicit lower bounds for the constants $θ(k,\varepsilon,n)$.

math.PR↗

arXiv2014

On colorings of variable words

In this note, we prove that the base case of the Graham--Rothschild Theorem, i.e., the one that considers colorings of the ($1$-dimensional) variable words, admits bounds in the class $\mathcal{E}^5$ of Grzegorczyk's hierarchy.

math.CO↗

arXiv2014

On the structure of the set of higher order spreading models

We generalize some results concerning the classical notion of a spreading model for the spreading models of order $ξ$. Among them, we prove that the set $SM_ξ^w(X)$ of the $ξ$-order spreading models of a Banach space $X$ generated by subordinated weakly null $\mathcal{F}$-sequences endowed with the pre-partial order of domination is a semi-lattice. Moreover, if $SM_ξ^w(X)$ contains an increasing sequence of length $ω$ then it contains an increasing sequence of length $ω_1$. Finally, if $SM_ξ^w(X)$ is uncountable, then it contains an antichain of size the continuum.

math.FA↗

arXiv2014

A disjoint union theorem for trees

We prove an infinitary disjoint union theorem for level products of trees. To implement the proof we develop a Hales-Jewett type result for words indexed by a level product of trees.

math.CO↗

arXiv2013

Combinatorial Structures on van der Waerden sets

In this paper we provide two results. The first one consists an infinitary version of the Furstenberg-Weiss Theorem. More precisely we show that every subset $A$ of a homogeneous tree $T$ such that $\frac{|A\cap T(n)|}{|T(n)|}\geqδ$, where T(n) denotes the $n$-th level of $T$, for all $n$ in a van der Waerden set, for some positive real $δ$, contains a strong subtree having a level sets which forms a van der Waerden set. The second result is the following. For every sequence $(m_q)_{q}$ of positive integers and for every real $0<δ\\leq1$, there exists a sequence $(n_q)_{q}$ of positive integers such that for every $D\subseteq \bigcup_k\prod_{q=0}^{k-1}[n_q]$ satisfying $$\frac{\big{|}D\cap \prod_{q=0}^{k-1} [n_q]\big{|}}{\prod_{q=0}^{k-1}n_q}\geqδ$$ for every $k$ in a van der Waerden set, there is a sequence $(J_q)_{q}$, where $J_q$ is an arithmetic progression of length $m_q$ contained in $[n_q]$ for all $q$, such that $\prod_{q=0}^{k-1}J_q\subseteq D$ for every $k$ in a van der Waerden set. Moreover, working in an abstract setting, we obtain $J_q$ to be any configuration of natural numbers that can be found in an arbitrary set of positive density.

math.CO↗

arXiv2013

Measurable events indexed by products of trees

A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\meg 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. A vector homogeneous tree $\mathbf{T}$ is a finite sequence $(T_1,...,T_d)$ of homogeneous trees and its level product $\otimes\mathbf{T}$ is the subset of the cartesian product $T_1\times ...\times T_d$ consisting of all finite sequences $(t_1,...,t_d)$ of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product $\otimes\mathbf{T}$ of a vector homogeneous tree $\mathbf{T}$. We show that, by refining the index set to the level product $\otimes\mathbf{S}$ of a vector strong subtree $\bfcs$ of $\mathbf{S}$, such families of events become highly correlated. An analogue of Lebesgue's density Theorem is also established which can be considered as the "probabilistic" version of the density Halpern--Läuchli Theorem.

math.CO↗

arXiv2013

A simple proof of the density Hales-Jewett theorem

We give a purely combinatorial proof of the density Hales--Jewett Theorem that is modeled after Polymath's proof but is significantly simpler. In particular, we avoid the use of the equal-slices measure and work exclusively with the uniform measure.

math.CO↗

arXiv2012

Subsets of Products of Finite Sets of Positive Upper Density

In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<δ\leqslant1$ there is a sequence $(n_q)_{q}$ of positive integers such that for every sequence $(H_q)_{q}$ of finite sets such that $|H_q|=n_q$ for every $q\in\mathbb{N}$ and for every $D\subseteq \bigcup_k\prod_{q=0}^{k-1}H_q$ with the property that $$\limsup_k \frac{|D\cap \prod_{q=0}^{k-1} H_q|}{|\prod_{q=0}^{k-1}H_q|}\geqslantδ$$ there is a sequence $(J_q)_{q}$, where $J_q\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that $\prod_{q=0}^{k-1}J_q\subseteq D$ for infinitely many $k.$ This gives us a density version of a well-known Ramsey-theoretic result. We also give some estimates on the sequence $(n_q)_{q}$ in terms of the sequence of $(m_q)_{q}$.

math.CO↗

arXiv2012

Oscilation stability for continuous monotone surjections

We prove that for every integer $b\geqslant 2$ and positive real $\varepsilon$ there exists a finite number $t$ such that for every finite coloring of the nondecreasing surjections from $b^ω$ onto $b^ω$, there exist $t$ many colors such that their $\varepsilon$-fattening contains a cube.

math.CO↗

arXiv2012

Measurable events indexed by trees

A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees. Precisely, we show that for every integer $b\geq 2$ and every integer $n\geq 1$ there exists an integer $q(b,n)$ with the following property. If $T$ is a homogeneous tree with branching number $b$ and $\{A_t:t\in T\}$ is a family of measurable events in a probability space $(Ω,Σ,μ)$ satisfying $μ(A_t)\geqε>0$ for every $t\in T$, then for every $0<θ<ε$ there exists a strong subtree $S$ of $T$ of infinite height such that for every non-empty finite subset $F$ of $S$ of cardinality $n$ we have \[ μ\Big(\bigcap_{t\in F} A_t\Big) \meg θ^{q(b,n)}. \] In fact, we can take $q(b,n)= \big((2^b-1)^{2n-1}-1\big)\cdot(2^b-2)^{-1}$. A finite version of this result is also obtained.

math.CO↗

arXiv2012

Dense subsets of products of finite trees

We prove a "uniform" version of the finite density Halpern-Läuchli Theorem. Specifically, we say that a tree $T$ is homogeneous if it is uniquely rooted and there is an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We show the following. For every integer $d\geq 1$, every $b_1,...,b_d\in\mathbb{N}$ with $b_i\geq 2$ for all $i\in\{1,...,d\}$, every integer $k\meg 1$ and every real $0<ε\leq 1$ there exists an integer $N$ with the following property. If $(T_1,...,T_d)$ are homogeneous trees such that the branching number of $T_i$ is $b_i$ for all $i\in\{1,...,d\}$, $L$ is a finite subset of $\mathbb{N}$ of cardinality at least $N$ and $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying \[|D\cap \big(T_1(n)\times ...\times T_d(n)\big)| \geq ε|T_1(n)\times ...\times T_d(n)|\] for every $n\in L$, then there exist strong subtrees $(S_1,...,S_d)$ of $(T_1,...,T_d)$ of height $k$ and with common level set such that the level product of $(S_1,...,S_d)$ is contained in $D$. The least integer $N$ with this property will be denoted by $UDHL(b_1,...,b_d|k,ε)$. The main point is that the result is independent of the position of the finite set $L$. The proof is based on a density increment strategy and gives explicit upper bounds for the numbers $UDHL(b_1,...,b_d|k,ε)$.

math.CO↗