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For r ∈ N\{0} an r-edge coloring of a graph or hypergraph G = (V, E) is a map c : E → {0,. .. , r −1}. Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that • every r-edge colored complete k-uniform hypergraph on N can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a k-uniform(More)
Let K ⊂ R d be a self-similar or self-affine set, let µ be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpi´nski sponge) we(More)
In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdor dimension. For a compact metric space K let dimH K and dimtH K denote its Hausdor and topological Hausdor dimension, respectively. We proved that this new dimension describes the Hausdor dimension of the level sets of the generic continuous(More)
In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class ξ functions, and generalize most of the results from the Baire class 1 case. We also show(More)
Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure µ on G such that µ(B + g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old(More)
A subset X of a Polish group G is called Haar null if there exists a Borel set B ⊃ X and Borel probability measure µ on G such that µ(gBh) = 0 for every g, h ∈ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure µ such that µ(X + t) = 0 for every t ∈ R. This answers a question from David Fremlin's problem list(More)
We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in ZF C and even in the theory ZF C + c = ω 2 if the number of pieces can be(More)
In the 1970s M. Laczkovich posed the following problem: Let B1(X) denote the set of Baire class 1 functions defined on a Polish space X equipped with the pointwise ordering. Characterize the order types of the linearly ordered subsets of B1(X). The main result of the present paper is a complete solution to this problem. We prove that a linear order is(More)