arXiv subjects

V. Farmaki

Publications and source records attributed to V. Farmaki.

3 recordsLinked to original sources

arXiv2005

Schreier Sets in Ramsey Theory

We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generating from k-tuples of words for any fixed natural number k) into a partition theorem about combinatorial subspaces (generating from Schreier-type sets of words of order any fixed countable ordinal). Furthermore, as a result we obtain a strengthening of Carlson's infinitary Nash-Williams type (and Ellentuck type) partition theorem about infinite sequences of variable words into a theorem, in which an infinite sequence of variable words and a binary partition of all the finite sequences of words, one of whose components is, in addition, a tree, are assumed, concluding that all the Schreier-type finite reductions of an infinite reduction of the given sequence have a behavior determined by the Cantor-Bendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.

math.CO↗

arXiv2004

Block Combinatorics

In this paper we extend the block combinatorics partition theorems of Hindman and Milliken in the setting of the recursive system of the block Schreier families (B^xi) consisting of families defined for every countable ordinal xi. Results contain (a) a block partition Ramsey theorem for every countable ordinal xi (Hindman's theorem corresponding to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.

math.CO↗

arXiv1998

Ramsey dichotomies with ordinal index

A system of uniform families on an infinite subset $M$ of $\nn$ is a collection $(\cca_ξ)_{ξ<ω_1}$ of families of finite subsets of $\nn$ (where, $\cca_k$ consists of all $k$--element subset of $M$, for $k\in \nn$) with the properties that each $\cca_ξ$ is thin (i.e. it does not contain proper initial segments of any of its element) and the Cantor--Bendixson index, defined for $\cca_ξ$, is equal to $ξ+1$ and stable when we restrict ourselves to any subset of $M$. We indicate how to extend the generalized Schreier families to a system of uniform families. Using that notion we establish the correct (countable) ordinal index generalization of the classical Ramsey theorem (which corresponds to the finite ordinal indices).

math.LO↗