András Máthé

Self-similar and self-affine sets: measure of the intersection of two copies

Let Kd be a self-similar or self-affine set and let μ be a self-similar or self-affine measure on it. Let

The Angel of Power 2 Wins

We solve Conways 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 to a square where he could have already landed at a previous time. We turn this observation around and prove that we may suppose without loss of generality that the Devil never eats a square where the Angel could have already jumped. Then we give a simple winning strategy for the Angel.

Poset limits can be totally ordered

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 each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probability space admits a measure-preserving and almost order-preserving map to the unit interval.

Hausdorff Dimension of Metric Spaces and Lipschitz Maps onto Cubes

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 always be mapped onto a k-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application, we essentially answer a question of Urbanski by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset A of a complete separable metric space is ⌊dimHA⌋ if dimHA is finite but not an integer, dimHA or dimHA−1 if dimHA is an integer and at least ω0 if Graphic.

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lovasz 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 sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.

Cliques in dense inhomogeneous random graphs

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant math formula of the Erdős–Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of math formula for each fixed n, and give examples of graphons for which math formula exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite.

Answer to a question of Kolmogorov

More than 80 years ago Kolmogorov asked the following question. Let

A proof of the dense version of Lovász conjecture

E\subseteq \mathbb{R}^{2}

How large dimension guarantees a given angle

be a measurable set with

Sets of large dimension not containing polynomial configurations

\lambda^{2}(E) 0