Recent developments in graph Ramsey theory

  title={Recent developments in graph Ramsey theory},
  author={David Conlon and Jacob Fox and Benny Sudakov},
  booktitle={Surveys in Combinatorics},
Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic copy of $H$. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of… 
A Survey of Hypergraph Ramsey Problems
The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple
Three early problems on size Ramsey numbers
The size Ramsey number of a graph H is defined as the minimum number of edges in a graph G such that there is a monochromatic copy of H in every two-coloring of E(G). The size Ramsey number was
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
The minimum degree of minimal Ramsey graphs for cliques
We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erdős and Lovasz in 1976, which is defined as the smallest minimum degree of a graph $G$ such
Ramsey numbers of cycles versus general graphs
The Ramsey number R(F,H) is the minimum number N such that any N -vertex graph either contains a copy of F or its complement contains H . Burr in 1981 proved a pleasingly general result that for any
Density of monochromatic infinite subgraphs II
In 1967, Gerencser and Gyarfas proved the following seminal result in graph-Ramsey theory: every 2-colored $K_n$ contains a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best
Anti-Ramsey multiplicities
Anti-Ramsey multiplicity is investigated for several families of graphs, finding classes of graphs which are either anti-common or not, and the rainbow equivalent of Sidorenko's conjecture, that all bipartite graphs are anti- common, is false.
Threshold Ramsey multiplicity for odd cycles
The Ramsey number r(H) of a graph H is the minimum n such that any two-coloring of the edges of the complete graph Kn contains a monochromatic copy of H. The threshold Ramsey multiplicity m(H) is
A Note on Induced Ramsey Numbers
The induced Ramsey number r ind(F) of a k-uniform hypergraph F is the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that every two-coloring of the edges


A conjecture of Erdős on graph Ramsey numbers
The Ramsey number of dense graphs
The Ramsey number r(H) of a graph H is the smallest number n such that, in any two‐colouring of the edges of Kn, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t
On the Ramsey Number of Sparse 3-Graphs
This work considers a hypergraph generalization of a conjecture of Burr and Erdős concerning the Ramsey number of graphs with bounded degree and derives the analogous result for 3-uniform hypergraphs.
Induced Ramsey Numbers
The induced Ramsey number of pairs of graphs (G, H) is investigated to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copies of H arises.
Ramsey-Type Problem for an Almost Monochromatic K4
It is proved that there is a constant c such that every-edge-coloring of the complete graph K_n with n contains a K_4 whose edges receive at most two colors, which is the first exponential bound for this problem.
Generalizations of a Ramsey-theoretic result of chvátal
The results proved all support the conjecture that any large graph that is sufficiently sparse, in the appropriate sense, is k-good, and such a T is called kgood.
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
On two problems in graph Ramsey theory
This work improves the upper bound on the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2cnlogn, and moves a step closer to proving this conjecture.
Ramsey-type Theorems with Forbidden Subgraphs
This work answers the question in the affirmative that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least and establishes several Ramsey type results for tournaments.
Ramsey-type theorems