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 1Γ(k+1) is a subset of a periodic k-sum-free set a
Sum-free sets and Ramsey numbers
β Scribed by D. Hanson
- Publisher
- Elsevier Science
- Year
- 1976
- Tongue
- English
- Weight
- 451 KB
- Volume
- 14
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
β¦ Synopsis
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.
π SIMILAR VOLUMES
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.
Simple proofs are given for three infinite classes of zero-sum Ramsey numbers modulo 3: r(K n , Z 3 )=n+3 for n#1, 4 (mod 9) and r(K n , Z 3 )=n+4 for n#0 (mod 9).
Caro, Y., On zero-sum Ramsey numbers--stars, Discrete Mathematics 104 (1992) l-6. Let n 3 k 2 2 be positive integers, k ( n. Let H, be the cyclic group of order k. Denote by R(K,,,> Z,) the minimal integer t such that for every &-coloring of the edges of K,, (i.e., a function c : E(K,)+ hk), there i
The Ramsey number r=r(G1-GZ-...-G,,,,H1-Hz-...-Hn) denotes the smallest r such that every 2-coloring of the edges of the complete graph K, contains a subgraph Gi with all edges of one color, or a subgraph Hi with all edges of a second color. These Ramsey numbers are determined for all sets of graph