Dense subsets of products of finite trees

We prove a "uniform" version of the finite density Halpern-L\"{a}uchli 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 $0<\epsilon\leq 1$ 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 $$|D\cap \big(T_1(n)\times ...\times T_d(n)\big)| \geq \epsilon |T_1(n)\times ...\times T_d(n)|$$ 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 $UDHL(b_1,...,b_d|k,\epsilon)$. 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 $UDHL(b_1,...,b_d|k,\epsilon)$.Comment: 36 pages, no figures; International Mathematics Research Notices, to appea

Remarks on a Ramsey theory for trees

Extending Furstenberg's ergodic theoretic proof for Szemer\'edi's theorem on arithmetic progressions, Furstenberg and Weiss (2003) proved the following qualitative result. For every d and k, there exists an integer N such that no matter how we color the vertices of a complete binary tree T_N of depth N with k colors, we can find a monochromatic replica of T_d in T_N such that (1) all vertices at the same level in T_d are mapped into vertices at the same level in T_N; (2) if a vertex x of T_d is mapped into a vertex y in T_N, then the two children of x are mapped into descendants of the the two children of y in T_N, respectively; and 3 the levels occupied by this replica form an arithmetic progression. This result and its density versions imply van der Waerden's and Szemer\'edi's theorems, and laid the foundations of a new Ramsey theory for trees. Using simple counting arguments and a randomized coloring algorithm called random split, we prove the following related result. Let N=N(d,k) denote the smallest positive integer such that no matter how we color the vertices of a complete binary tree T_N of depth N with k colors, we can find a monochromatic replica of T_d in T_N which satisfies properties (1) and (2) above. Then we have N(d,k)=\Theta(dk\log k). We also prove a density version of this result, which, combined with Szemer\'edi's theorem, provides a very short combinatorial proof of a quantitative version of the Furstenberg-Weiss theorem.Comment: 10 pages 1 figur

Trees and Markov convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors

A density version of the Carlson--Simpson theorem

We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying $$\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0$$ there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set $$\{c\}\cup \big\{c^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_n(a_n) : n\in\mathbb{N} \ \text{ and } \ a_0,...,a_n\in [k]\big\}$$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.Comment: 73 pages, no figure

Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory

The aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory