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Sum-free sets and ramsey numbers II

✍ Scribed by D. Hanson; J. Hanson

Book ID: 107748241
Publisher: Elsevier Science
Year: 1977
Tongue: English
Weight: 151 KB
Volume: 20
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(77)90068-1

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Sum-free sets and Ramsey numbers

✍ D. Hanson 📂 Article 📅 1976 🏛 Elsevier Science 🌐 English ⚖ 451 KB

in this note we obtain new tower bounds for the Ramsey numbers R(5,S) and R(5,6). The methrld is based on computational results of partitioning the integers into sum-free sets. WC obtain R(S, 5) > 42 and R(5,6) 2 53.

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A subset of the natural numbers is k-sum-free if it contains no solutions of the equation x 1 + } } } +x k = y, and strongly k-sum-free when it is l-sum-free for every l=2, ..., k. It is shown that every k-sum-free set with upper density larger than 1Â(k+1) is a subset of a periodic k-sum-free set a

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A counting argument is developed and divisibility properties of the binomial coefficients are combined to prove, among other results, that where K n , resp. K k n , is the complete, resp. complete k-uniform, hypergaph and R(K n , Z p ), R(K k n , Z 2 ) are the corresponding zero-sum Ramsey numbers.

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✍ Calkin, N. J. 📂 Article 📅 1990 🏛 Oxford University Press 🌐 English ⚖ 84 KB

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✍ Wong, Tony W.H. 📂 Article 📅 2014 🏛 Elsevier Science 🌐 English ⚖ 316 KB