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# An uncountably chromatic triple system

Acta Mathematica Hungarica
Author: P. Komjáth

## Abstract

It is consistent that there exists an uncountably chromatic triple system which does not contain two triples with two common points or circuits of lengths 3, 5.

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# Obligatory subsystems of triple systems

Acta Mathematica Hungarica
Authors: A. Hajnal and P. Komjáth

## Abstract

We determine a class of triple systems such that each must occur in a triple system with uncountable chromatic number that omits
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{T}_0$$ \end{document}
(the unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also show that consistently there are two finite triple systems such that they can separately be omitted by uncountably chromatic triple systems but not both.
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# Conflict free colorings of (strongly) almost disjoint set-systems

Acta Mathematica Hungarica
Authors: András Hajnal, István Juhász, Lajos Soukup, and Zoltán Szentmiklóssy

## Abstract is called a conflict free coloring of the set-system (with ρ colors) if The conflict free chromatic number of is the smallest ρ for which admits a conflict free coloring with ρ colors. is a (λ,κ,μ)-system if , |A|=κ for all , and is μ-almost disjoint, i.e. |AA′|<μ for distinct . Our aim here is to study for λκμ, actually restricting ourselves to λω and μω.

For instance, we prove that

• for any limit cardinal κ (or κ=ω) and integers n≧0, k>0, GCH implies • if λκω>d>1, then λ<κ +ω implies and λ≧ℶω(κ) implies ;

• GCH implies for λκω 2 and V=L implies for λκω 1;

• the existence of a supercompact cardinal implies the consistency of GCH plus and for 2≦nω;

• CH implies , while implies .

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# A note on the Engelking-Karłowicz theorem

Acta Mathematica Hungarica
Authors: U. Abraham and Y. Yin

## Abstract

We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Karłowicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some aXY the order-type of aX is different from that of aY. In addition to the chromatic number x(G) of these graphs we study χκ(G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.

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# L-spaces and the P-ideal dichotomy

Acta Mathematica Hungarica
Authors: H. Mildenberger and L. Zdomskyy

## Abstract

We extend a theorem of Todorčević: Under the assumption (
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) (see Definition 1.11),
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\boxtimes \left\{ \begin{gathered} any regular space Z with countable tightness such that \hfill \\ Z^n is Lindel\ddot of for all n \in \omega has no L - subspace. \hfill \\ \end{gathered} \right.$$ \end{document}
We assume
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> ω 1 and a weak form of Abraham and Todorčević’s P-ideal dichotomy instead and get the same conclusion. Then we show that
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> ω 1 and the dichotomy principle for P-ideals that have at most ℵ1 generators together with ⊠ do not imply that every Aronszajn tree is special, and hence do not imply (ie1-4). So we really extended the mentioned theorem.
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# Hereditarily Lindelöf spaces of singular density

Studia Scientiarum Mathematicarum Hungarica
Authors: István Juhász and Saharon Shelah

A cardinal λ is called ω -inaccessible if for all µ < λ we have µ ω < λ . We show that for every ω -inaccessible cardinal λ there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelöf regular space of density λ . This extends an analogous earlier result of ours that only worked for regular λ .

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# Gerlits and function spaces

Studia Scientiarum Mathematicarum Hungarica
Author: Marion Scheepers

János Gerlits died unexpectedly in 2008. In this paper we attempt to make a small tribute to his very powerful mathematical legacy by describing the emerging impact of two ideas, γ-spaces and Gerlits-Nagy spaces, from  and . Restricted access