• Corpus ID: 233864385

Ramsey numbers of sparse digraphs

  title={Ramsey numbers of sparse digraphs},
  author={Jacob Fox and Xiaoyu He and Yuval Wigderson},
Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr–Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If H is an acyclic digraph, the oriented Ramsey number of H , denoted r1(H), is the least N such that every tournament on N vertices contains a copy of H . We show that for any… 

Figures from this paper

On 1-Subdivisions of Transitive Tournaments

The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the

Powers of paths and cycles in tournaments

We show that for every positive integer k, any tournament can be partitioned into at most 2 k-th powers of paths. This result is tight up to the exponential constant. Moreover, we prove that for




If G and H are graphs (which will mean finite, with no loops or parallel lines), define the Ramsey number r(G, H) to be the least number p such that if the lines of the complete graph Kp are colored

Ordered Ramsey numbers

Directed Ramsey number for trees

Ramsey numbers of degenerate graphs

A graph is $d$-degenerate if all its subgraphs have a vertex of degree at most $d$. We prove that there exists a constant $c$ such that for all natural numbers $d$ and $r$, every $d$-degenerate graph

Powers of paths in tournaments

It is proved that every tournament contains the k-th power of a directed path of linear length, and a complete solution for this problem when k=2 is given, showing that there is always a square of adirected path of length , which is best possible.

Paths with many shortcuts in tournaments

Diagonal Ramsey via effective quasirandomness

We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey

Models of random regular graphs

This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results. Related results on asymptotic enumeration are

Graphs and Geometry

The Voting Problem