Ramsey class

In the area of mathematics known as Ramsey theory, a Ramsey class^[1] is one which satisfies a generalization of Ramsey's theorem.

Suppose ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ are structures and ${\displaystyle k}$ is a positive integer. We denote by ${\displaystyle {\binom {B}{A}}}$ the set of all subobjects ${\displaystyle A'}$ of ${\displaystyle B}$ which are isomorphic to ${\displaystyle A}$. We further denote by ${\displaystyle C\rightarrow (B)_{k}^{A}}$ the property that for all partitions ${\displaystyle X_{1}\cup X_{2}\cup \dots \cup X_{k}}$ of ${\displaystyle {\binom {C}{A}}}$ there exists a ${\displaystyle B'\in {\binom {C}{B}}}$ and an ${\displaystyle 1\leq i\leq k}$ such that ${\displaystyle {\binom {B'}{A}}\subseteq X_{i}}$.

Suppose ${\displaystyle K}$ is a class of structures closed under isomorphism and substructures. We say the class ${\displaystyle K}$ has the A-Ramsey property if for ever positive integer ${\displaystyle k}$ and for every ${\displaystyle B\in K}$ there is a ${\displaystyle C\in K}$ such that ${\displaystyle C\rightarrow (B)_{k}^{A}}$ holds. If ${\displaystyle K}$ has the ${\displaystyle A}$-Ramsey property for all ${\displaystyle A\in K}$ then we say ${\displaystyle K}$ is a Ramsey class.

Ramsey's theorem is equivalent to the statement that the class of all finite sets is a Ramsey class.

^{[2] [3]}

References

1. Nešetřil, Jaroslav (2016-06-14). "All the Ramsey Classes - צילום הרצאות סטודיו האנה בי - YouTube". www.youtube.com. Tel Aviv University. Retrieved 4 November 2020.
2. Bodirsky, Manuel (27 May 2015). "Ramsey Classes: Examples and Constructions". arXiv:1502.05146 [math.CO].
3. Hubička, Jan; Nešetřil, Jaroslav (November 2019). "All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)". Advances in Mathematics. 356: 106791. arXiv:1606.07979. doi:10.1016/j.aim.2019.106791. S2CID 7750570.