Related papers: List Ramsey numbers
We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of…
We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's…
An r-book of size q is a union of q (r+1)-cliques sharing a common r-clique. We find exactly the Ramsey number of a p-clique versus r-books of sufficiently large size. Furthermore, we find asymptotically the Ramsey number of any fixed…
Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic…
Let $r_k(s, e; t)$ denote the smallest $N$ such that any red/blue edge coloring of the complete $k$-uniform hypergraph on $N$ vertices contains either $e$ red edges among some $s$ vertices, or a blue clique of size $t$. Erd\H os and Hajnal…
Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdos and Fred Galvin proved that for each coloring f, there is an infinite set that is "packed together" which…
Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed…
In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph…
The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one…
Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A. %Ruci\'{n}ski, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of…
The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…
A construction described by the current author (2017) uses two linear prototypes to build a compound graph with Ramsey properties inherited from the prototype graphs. The resulting graph is linear; and cyclic if both prototypes are cyclic.…
A graph $G$ is semilinear of complexity $t$ if the vertices of $G$ are elements of $\mathbb{R}^{d}$ for some $d\in\mathbb{Z}^{+}$, and the edges of $G$ are defined by the sign patterns of $t$ linear functions…
We exhibit a family of $3$-uniform hypergraphs with the property that their $2$-colour Ramsey numbers grow polynomially in the number of vertices, while their $4$-colour Ramsey numbers grow exponentially. This is the first example of a…
For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or…
An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly…
The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with…
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable reducibility. We proceed to a systematic…
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset…
We show that canonical Ramsey numbers for partite hypergraphs grow single exponentially for any fixed uniformity.