Hypergraph Packing and Sparse Bipartite Ramsey Numbers
نویسنده
چکیده
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 and Taraz.
منابع مشابه
Zarankiewicz Numbers and Bipartite Ramsey Numbers
The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...
متن کاملDiagonal Forms, Linear Algebraic Methods and Ramsey-Type Problems
This thesis focuses mainly on linear algebraic aspects of combinatorics. Let Nt(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of Nt(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H. As a continu...
متن کاملSparse hypergraphs with low independence number
Let K (3) 4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erdős, Komlós, and Szemerédi (1981) asked if there is a function ω(d) → ∞ such that every 3-uniform, K (3) 4 -free hypergraph H with N vertices and average degree d has independence number at least N d1/2 ω(d). We answer this question by constructing a 3-uniform, K (3) 4 -free hypergraph with independence number at most ...
متن کاملRamsey numbers of sparse hypergraphs
We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree has Ramsey number at most c( , k)n, for an appropriate constant c( , k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self-contained proof which uses new techniques developed recently by th...
متن کاملCombinatorial Algorithms for Parallel Sparse Matrix Distributions
Combinatorial algorithms have long played a crucial enabling role in parallel computing. An important problem in sparse matrix computations is how to distribute the sparse matrix and vectors among processors. We review graph, bipartite graph, and hypergraph models for both 1d (row or column) distributions and 2d distributions. A valuable tool is hypergraph partitioning. We present results using...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Combinatorics, Probability & Computing
دوره 18 شماره
صفحات -
تاریخ انتشار 2009