@@ -6,11 +6,9 @@ The fundamental principle of Ramsey theory can be described as follows: every la

In Graph Ramsey Theory the above principle is usually presented through the discussion of edge colourings: given any two graphs $H$ and $G$, we say that $G$ is a Ramsey graph for $H$, denoted by $G\rightarrow H$, if every colouring of the edges of $G$ with only two colours leads to a monochromatic subgraph of $G$ which is isomorphic to $H$. The fundamental result of Ramsey [Ra1930] then states that for every $H$ there exists a graph $G$ such that $G\rightarrow H$ holds. For instance, denote with $K_n$ a complete graph on $n$ vertices, then it is an easy exercise to show that $K_6\rightarrow K_3$.

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Obviously, a natural question to ask is how many vertices a graph needs in order to be Ramsey for another graph $H$. This leads to the famous Ramsey number $r(H)$, which turns out to be incredibely hard to determine even for the case when $H=K_5$. Indeed, for general $n$, the best known bounds on $r(K_n)$ are of the form