## 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.

## Abstract

## Abstract

*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.

*λ*,

*κ*,

*μ*)-system if , |

*A*|=

*κ*for all , and is

*μ*-almost disjoint, i.e. |

*A*∩

*A*′|<

*μ*for distinct . Our aim here is to study for

*λ*≧

*κ*≧

*μ*, actually restricting ourselves to

*λ*≧

*ω*and

*μ*≦

*ω*.

For instance, we prove that

*κ*(or

*κ*=

*ω*) and integers

*n*≧0,

*k*>0, GCH implies

• if *λ*≧*κ*≧*ω*>*d*>1, then *λ*<*κ*
^{+ω} implies and *λ*≧ℶ_{ω}(*κ*) implies ;

• GCH implies for *λ*≧*κ*≧*ω*
_{2} and *V*=*L* implies for *λ*≧*κ*≧*ω*
_{1};

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

• CH implies , while implies .

## 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 *a* ∈ *X* ∩ *Y* the order-type of *a* ∩ *X* is different from that of *a* ∩ *Y*.
In addition to the chromatic number *x*(*G*) of these graphs we study χ_{κ}(*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.

## Abstract

*ω*

_{1}and a weak form of Abraham and Todorčević’s

*P*-ideal dichotomy instead and get the same conclusion. Then we show that

*ω*

_{1}and the dichotomy principle for

*P*-ideals that have at most ℵ

_{1}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.

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*
.

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 [19] and [20].