The Ramsey Number for Hypergraph Cycles Ii
نویسنده
چکیده
Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.
منابع مشابه
The 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 کاملThe Lifting of Graphs to 3-uniform Hypergraphs and Some Applications to Hypergraph Ramsey Theory
Given a simple graph Γ, we describe a “lifting” to a 3-uniform hypergraph φ(Γ) that sends the complement of Γ to the complement of φ(Γ). We consider the effects of this lifting on cycles, complete subhypergraphs, and complete subhypergraphs missing a single hyperedge. Our results lead to natural lower bounds for some hypergraph Ramsey numbers.
متن کاملThe Ramsey Number for 3-Uniform Tight Hypergraph Cycles
Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...
متن کاملThe 3-colored Ramsey number for a 3-uniform loose path of length 3
The values of hypergraph 2-color Ramsey numbers for loose cycles and paths have already been determined. The only known value for more than 2 colors is R(C 3 ; 3) = 8, where C 3 3 is a 3-uniform loose cycle of length 3. Here we determine that R(P 3 3 ; 3) = 9, where P 3 3 is a 3-uniform loose path of length 3. Our proof relies on the determination of the Turán number ex3(9;P 3 3 ). We also find...
متن کاملGeneralized 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 کامل