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Ideal version of Ramsey's theorem

Rafał Filipów, Nikodem Mrożek, Ireneusz Recław, Piotr Szuca (2011)

Czechoslovak Mathematical Journal

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....

Identities arising from higher-order Daehee polynomial bases

Dae San Kim, Taekyun Kim (2015)

Open Mathematics

Here we will derive formulas for expressing any polynomial as linear combinations of two kinds of higherorder Daehee polynomial basis. Then we will apply these formulas to certain polynomials in order to get new and interesting identities involving higher-order Daehee polynomials of the first kind and of the second kind.

Increasing integer sequences and Goldbach's conjecture

Mauro Torelli (2006)

RAIRO - Theoretical Informatics and Applications

Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.

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