Márton Elekes

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Let X be a set, κ be a cardinal number and let H be a family of subsets of X which covers each x ∈ X at least κ times. What assumptions can ensure that H can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover H: among other situations, we consider covers of topological spaces by(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)
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 kuniform(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)
Let K ⊂ R 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. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpiński sponge) we prove(More)
We investigate the existence of well-ordered sequences of Baire 1 functions on separable metric spaces. Any set F of real valued functions defined on an arbitrary set X is partially ordered by the pointwise order, that is f ≤ g iff f(x) ≤ g(x) for all x ∈ X. In other words put f < g iff f(x) ≤ g(x) for all x ∈ X and f(x) 6= g(x) for at least one x ∈ X. Our(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 ZFC and even in the theory ZFC + c = ω2 if the number of pieces can be uncountable(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)