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Abstract. We first formulate several "combinatorial principles" concerning κ × ω matrices of subsets of ω and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any ground model V , provided that the parameter κ is an ω-inaccessible regular cardinal in V .Then in section 4 we present a large number of applications of these principles, mainly to topology. Some of these consequences had been established earlier in generic extensions obtained by adding ω 2 Cohen reals to ground models satisfying CH, mostly for the case κ = ω 2 .

An r-edge coloring of a graph or hypergraph G " pV, Eq is a map c : E Ñ t0, . . . , r1u. Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that• the vertex set of every r-edge colored countably infinite complete k-uniform hypergraph can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a k-uniform hypergraph is a sequence of distinct vertices such that every set of k consecutive vertices forms an edge);• for all natural numbers r and k there is a natural number M such that the vertex set of every r-edge colored countably infinite complete graph can be partitioned into M monochromatic k th powers of paths apart from a finite set

Abstract.We prove that if k is an uncountable regular cardinal and a compact T2 space X contains a free sequence of length k , then X also contains such a sequence that is convergent. This implies that under CH every nonfirst countable compact T2 space contains a convergent oi\-sequence and every compact T2 space with a small diagonal is metrizable, thus answering old questions raised by the first author and M. Husek, respectively. IntroductionWhile there are compact T2 spaces with no nontrivial convergent co-sequences, ßN being perhaps the best known such space, every infinite compact T2 space contains nontrivial convergent transfinite sequences. Indeed, as it is easy to see, if in such a space {Aa : a g cp} is a (strictly) decreasing sequence of closed sets with f]{Aa : a e cp} = {p}, a singleton, then for every choice of points qa G Aa\Aa+i , the sequence {qa : a G cp} convergences to p, provided that çp is a limit ordinal.Moreover, it is also easy to see that if the character of a point p in such a space is equal to k then a decreasing sequence of closed sets {Aa : a G k} does exist with f]{Aa : a e k} = {p}, hence p is the limit of a k-sequence. Consequently, since every infinite compact T2 space has a separable closed infinite subspace, in which then every point has character < c = 2W , we get that every such space has convergent sequences of length < c.On the other hand, it has also been known (see [BSV] or [vM]) that for example, ßN contains a convergent sequence of length co\ , without assuming CH, hence the natural question arises whether every infinite compact T2 space contains a convergent co or co\ sequence? This question was first formulated by Husek in the late seventies, and a related stronger problem was independently raised by the first author around the same time: Does every nonfirst countable compact T2 space contain a convergent co\ sequence?

Abstract. All spaces are assumed to be infinite Hausdorff spaces. We call a space anti-Urysohn (AU in short) iff any two non-emty regular closed sets in it intersect. We prove that• for every infinite cardinal κ there is a space of size κ in which fewer than cf (κ) many non-empty regular closed sets always intersect; • there is a locally countable AU space of size κ iff ω ≤ κ ≤ 2 c . A space with at least two non-isolated points is called strongly anti-Urysohn (SAU in short) iff any two infinite closed sets in it intersect. We prove that• if X is any SAU space then s ≤ |X| ≤ 2 2 c ;