The Ramsey Number for Hypergraph
نویسنده
چکیده
Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v3v4v5, v5v6v7, . . . , vn−1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.
منابع مشابه
Generalized Ramsey theorems for r-uniform hypergraphs
We show that several known Ramsey number inequalities can be extended to the setting of r-uniform hypergraphs. In particular, we extend Burr’s results on tree-star Ramsey numbers, providing exact evaluations for certain hypergraph Ramsey numbers. Then we turn our attention to proving a general multicolor hypergraph Ramsey number inequality from which generalizations of results due to Chvátal an...
متن کاملThe Ramsey Number of Loose Triangles and Quadrangles in Hypergraphs
Asymptotic values of hypergraph Ramsey numbers for loose cycles (and paths) were determined recently. Here we determine some of them exactly, for example the 2-color hypergraph Ramsey number of a k-uniform loose 3-cycle or 4-cycle: R(Ck 3 , Ck 3 ) = 3k − 2 and R(Ck 4 , Ck 4 ) = 4k − 3 (for k > 3). For more than 3 colors we could prove only that R(C3 3 , C3 3 , C3 3) = 8. Nevertheless, the r-col...
متن کاملA note on lower bounds for hypergraph Ramsey numbers
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that r3(l, l, l) ≥ 2 c log log l . The old bound, due to Erdős and Hajnal, was r3(l, l, l) ≥ 2 2 log2 .
متن کاملA survey of quantitative bounds for hypergraph Ramsey problems
The classical hypergraph Ramsey number rk(s, n) is the minimum N such that for every redblue coloring of the k-tuples of {1, . . . , N}, there are s integers such that every k-tuple among them is red, or n integers such that every k-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of r...
متن کاملNew lower bounds for hypergraph Ramsey numbers
The Ramsey number rk(s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1, . . . , N}, there are s integers such that every k-tuple among them is red, or n integers such that every k-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: r4(5, n) > 2 n log n and r4(6, n) > 2 2 1/5 , where c is an absolute positive...
متن کاملHypergraph Packing and Sparse Bipartite Ramsey Numbers
We prove that there exists a constant c such that, for any integer ∆, the Ramsey number of a bipartite graph on n vertices with maximum degree ∆ is less than 2n. A probabilistic argument due to Graham, Rödl and Ruciński implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of Rödl, Ruciński a...
متن کامل