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Relational quotients

Miodrag Sokić (2013)

Fundamenta Mathematicae

Let 𝒦 be a class of finite relational structures. We define ℰ𝒦 to be the class of finite relational structures A such that A/E ∈ 𝒦, where E is an equivalence relation defined on the structure A. Adding arbitrary linear orderings to structures from ℰ𝒦, we get the class 𝒪ℰ𝒦. If we add linear orderings to structures from ℰ𝒦 such that each E-equivalence class is an interval then we get the class 𝒞ℰ[𝒦*]. We provide a list of Fraïssé classes among ℰ𝒦, 𝒪ℰ𝒦 and 𝒞ℰ[𝒦*]. In addition, we classify...

Some combinatorics involving ξ-large sets

Teresa Bigorajska, Henryk Kotlarski (2002)

Fundamenta Mathematicae

We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε₀ and the notion of largeness is in the sense of Hardy hierarchy.

Some Ramsey type theorems for normed and quasinormed spaces

C. Henson, Nigel Kalton, N. Peck, Ignác Tereščák, Pavol Zlatoš (1997)

Studia Mathematica

We prove that every bounded, uniformly separated sequence in a normed space contains a “uniformly independent” subsequence (see definition); the constants involved do not depend on the sequence or the space. The finite version of this result is true for all quasinormed spaces. We give a counterexample to the infinite version in L p [ 0 , 1 ] for each 0 < p < 1. Some consequences for nonstandard topological vector spaces are derived.

Two variants of the size Ramsey number

Andrzej Kurek, Andrzej Ruciński (2005)

Discussiones Mathematicae Graph Theory

Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m ( G ) = m a x F ⊆ G | E ( F ) | / | V ( F ) | and define the Ramsey density m i n f ( H , r ) as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then m i n f ( H , r ) = ( R - 1 ) / 2 , where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál...

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