### $2kn-\left(\genfrac{}{}{0pt}{}{2k+1}{2}\right)$. A note on extremal combinatorics of cyclic split systems.

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T. Brown proved that whenever we color ${\mathcal{P}}_{f}\left(\mathbb{N}\right)$ (the set of finite subsets of natural numbers) with finitely many colors, we find a monochromatic structure, called an arithmetic copy of an $\omega $-forest. In this paper we show a canonical extension of this theorem; i.eẇhenever we color ${\mathcal{P}}_{f}\left(\mathbb{N}\right)$ with arbitrarily many colors, we find a canonically colored arithmetic copy of an $\omega $-forest. The five types of the canonical coloring are determined. This solves a problem of T. Brown.

A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals $1$ or $2$.

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\mathrm{lim}\phantom{\rule{4pt}{0ex}}{\mathrm{sup}}_{n\to \infty}|A\cap {\left[k\right]}^{n}|/{k}^{n}>0$ there exist a word $c$ over $k$ and a sequence $\left({w}_{n}\right)$ of left variable words over $k$ such that the set $c\cup \{{c}^{}{w}_{0}{\left({a}_{0}\right)}^{}..{.}^{}{w}_{n}\left({a}_{n}\right):n\in \mathbb{N}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}{a}_{0},...,{a}_{n}\in \left[k\right]\}$ 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.

We show that the Schreier sets ${S}_{\alpha}(\alpha <{\omega}_{1})$ have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite $M={\left({m}_{i}\right)}_{i=1}^{\infty}\subseteq \mathbb{N}$ such that ${S}_{\alpha}\left(M\right)={m}_{i}:i\in E:E\in {S}_{\alpha}\subseteq \mathcal{F}$, or there exist infinite $M={\left({m}_{i}\right)}_{i=1}^{\infty},N\subseteq \mathbb{N}$ such that $\mathcal{F}\left[N\right]\left(M\right)={m}_{i}:i\in F:F\in \mathcal{F}andF\subset N\subseteq {S}_{\alpha}$.

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${\left({\alpha}_{j}\right)}_{j=1}^{\infty}$ of scalars, there exists a subsequence ${\left({\alpha}_{{k}_{j}}\right)}_{j=1}^{\infty}$ such that either every subsequence of ${\left({\alpha}_{{k}_{j}}\right)}_{j=1}^{\infty}$ defines a universal series, or no subsequence of ${\left({\alpha}_{{k}_{j}}\right)}_{j=1}^{\infty}$ defines a universal series. In particular examples we decide which of the two cases holds.

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.

We investigate the extremal function $f(u,n)$ which, for a given finite sequence $u$ over $k$ symbols, is defined as the maximum length $m$ of a sequence $v={a}_{1}{a}_{2}...{a}_{m}$ of integers such that 1) $1\le {a}_{i}\le n$, 2) ${a}_{i}={a}_{j},i\ne j$ implies $|i-j|\ge k$ and 3) $v$ contains no subsequence of the type $u$. We prove that $f(u,n)$ is very near to be linear in $n$ for any fixed $u$ of length greater than 4, namely that $$f(u,n)=O\left(n{2}^{O\left(\alpha {\left(n\right)}^{\left|u\right|-4}\right)}\right).$$ Here $\left|u\right|$ is the length of $u$ and $\alpha \left(n\right)$ is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and [ASS] which...

Furstenberg's original Central Sets Theorem applied to central subsets of ℕ and finitely many specified sequences in ℤ. In this form it was already strong enough to derive some very strong combinatorial consequences, such as the fact that a central subset of ℕ contains solutions to all partition regular systems of homogeneous equations. Subsequently the Central Sets Theorem was extended to apply to arbitrary semigroups and countably many specified sequences. In this paper we derive a new version...