Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Clopen graphs

100%

Geschke S.

Fundamenta Mathematicae

2013

tom 220

nr 2

155-189

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their homogeneity numbers started in [11] and [9]. We show that clopen graphs on compact spaces have no infinite induced subgraphs that are 4-saturated. In particular, there are countably infinite graphs such as Rado's random graph that do not embed into any clopen graph on a compact space. Using similar methods, we show that the quasi-orders of clopen graphs on compact zero-dimensional metric spaces with topological or combinatorial embeddability are Tukey equivalent to $ω^{ω}$ with eventual domination. In particular, the dominating number 𝔡 is the least size of a family of clopen graphs on compact metric spaces such that every clopen graph on a compact zero-dimensional metric space embeds into a member of the family. We also show that there are ℵ₀-saturated clopen graphs on $ω^{ω}$, while no ℵ₁-saturated graph embeds into a clopen graph on a Polish space. There is, however, an ℵ₁-saturated $F_{σ}$ graph on $2^{ω}$.

Some combinatorial principles defined in terms of elementary submodels

63%

Fuchino S., Geschke S.

Fundamenta Mathematicae

2004

tom 181

nr 3

233-255

We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese-Nation property of 𝒫(ω) studied in [6] already follow from SEP. We show that it is consistent that SEP holds while 𝒫(ω) fails to have the (ℵ₁,ℵ ₀)-ideal property introduced in [2]. This answers a question addressed independently by Fuchino and by Kunen. We also consider some natural variants of SEP and show that certain changes in the definition of SEP do not lead to a different principle, answering a question of Blass.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Geschke S.

1 Fuchino S.

1 2013

1 2004

Wyniki wyszukiwania

Clopen graphs

Some combinatorial principles defined in terms of elementary submodels