We solve Conway’s Angel Problem by showing that the Angel of power 2 has a winning strategy. An old observation of Conway is that we may suppose without loss of generality that the Angel never jumps… (More)

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than k can be always mapped onto a k-dimensional cube by… (More)

We study the following two problems: (1) Given n ≥ 2 and α, how large Hausdorff dimension can a compact set A ⊂ R have if A does not contain three points that form an angle α? (2) Given α and δ, how… (More)

S. Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529–563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that… (More)

We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in R, for n ≥ 3, with non-empty interiors and of the same measure are equidecomposable using pieces that are… (More)

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (R, B, H) and (R, B, H) are not isomorphic if s 6= t, s, t ∈ [0, 1], where B is the… (More)

A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and… (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… (More)

We show that an analytic subset of the finite dimensional Euclidean space R is purely unrectifiable if and only if the image of any of its compact subsets under every local Lipschitz quotient… (More)