The Ramsey number r(H) of a graphH 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 vertices and density ρ, proving that r(H) ≤ 2 √ ρ . We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree ρt and the Ramsey number of random g...

We consider Fräıssé structures whose objects have finite big Ramsey degree and ask what consequences this has for the dynamics of the automorphism group. Motivated by a theorem of D. Devlin about the partition properties of the rationals, we define the notion of a big Ramsey structure, a single structure which codes the big Ramsey degrees of a given Fräıssé structure. This in turn leads to the ...

We show that if G is a Ramsey size-linear graph and x,y 2 V(G ) then if we add a sufficiently long path between x and y we obtain a new Ramsey size-linear graph. As a consequence we show that if G is any graph such that every cycle in G contains at least four consecutive vertices of degree 2 then G is Ramsey size-linear. ß 2002 John Wiley & Sons, Inc. J Graph Theory 39: 1–5, 2002

In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal w...

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers R(m,n) with m, n≥3, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers R(m,n). We show how the computation of R(m,n) can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evol...

The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω. We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In thi...