Viktor Harangi

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We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector(More)
A finite set H in Rd is called an acute set if any angle determined by three points of H is acute. We examine the maximal cardinality α(d) of a d-dimensional acute set. The exact value of α(d) is known only for d ≤ 3. For each d ≥ 4 we improve on the best known lower bound for α(d). We present different approaches. On one hand, we give a probabilistic proof(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 large Hausdorff dimension can a compact subset A of a Euclidean space have if A does not contain three points that form an angle in the δ-neighborhood of α?(More)
The thesis addresses problems from the field of geometric measure theory. It turns out<lb>that discrete methods can be used efficiently to solve these problems. Let us summarize<lb>the main results of the thesis.<lb>In Chapter 2 we investigate the following question proposed by Tamás Keleti. How<lb>large (in terms of Hausdorff dimension) can a compact set A(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 large Hausdorff dimension can a compact subset A of a Euclidean space have if A does not contain three points that form an angle in the δ-neighborhood of α?(More)
Consider a1, . . . , an ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of aj-periodic functions, i.e., f = f1+· · ·+fn with ∆aj f(x) := f(x+aj)−f(x) = 0. We show that f has such a decomposition if and only if for all partitions B1∪B2∪· · ·∪BN = {a1, . . . , an} with Bj consisting of commensurable elements with(More)
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