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A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

Acta Arithmetica

A geometric progression of length k and integer ratio is a set of numbers of the form a , a r , . . . , a r k - 1 for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ( a i ) i = 1 ∞ of positive real numbers with a₁ = 1 such that the set G ( k ) = ⋃ i = 1 ∞ ( a 2 i , a 2 i - 1 ] contains no geometric progression of length k and integer ratio. Moreover, G ( k ) is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...

A proof of menger's theorem by contraction

Frank Göring (2002)

Discussiones Mathematicae Graph Theory

A short proof of the classical theorem of Menger concerning the number of disjoint AB-paths of a finite graph for two subsets A and B of its vertex set is given. The main idea of the proof is to contract an edge of the graph.

A proof of the crossing number of K 3 , n in a surface

Pak Tung Ho (2007)

Discussiones Mathematicae Graph Theory

In this note we give a simple proof of a result of Richter and Siran by basic counting method, which says that the crossing number of K 3 , n in a surface with Euler genus ε is ⎣n/(2ε+2)⎦ n - (ε+1)(1+⎣n/(2ε+2)⎦).

A q-analogue of complete monotonicity

Anna Kula (2008)

Colloquium Mathematicae

The aim of this paper is to give a q-analogue for complete monotonicity. We apply a classical characterization of Hausdorff moment sequences in terms of positive definiteness and complete monotonicity, adapted to the q-situation. The method due to Maserick and Szafraniec that does not need moments turns out to be useful. A definition of a q-moment sequence appears as a by-product.

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