Ramsey class

{{Short description|Class satisfying a generalization of Ramsey's theorem}}

In the area of mathematics known as Ramsey theory, a Ramsey class{{cite web |last1=Nešetřil |first1=Jaroslav |title=All the Ramsey Classes - צילום הרצאות סטודיו האנה בי - YouTube |url=https://youtube.com/watch?v=_pfa5bogr8g |website=www.youtube.com |publisher=Tel Aviv University |accessdate=4 November 2020 |date=2016-06-14}} is one which satisfies a generalization of Ramsey's theorem.

Suppose $A$, $B$ and $C$ are structures and $k$ is a positive integer. We denote by $\backslash binom\{B\}\{A\}$ the set of all subobjects $A\text{'}$ of $B$ which are isomorphic to $A$. We further denote by $C\; \backslash rightarrow\; (B)^A\_k$ the property that for all partitions $X\_1\; \backslash cup\; X\_2\backslash cup\; \backslash dots\backslash cup\; X\_k$ of $\backslash binom\{C\}\{A\}$ there exists a $B\text{'}\; \backslash in\; \backslash binom\{C\}\{B\}$ and an $1\; \backslash leq\; i\; \backslash leq\; k$ such that $\backslash binom\{B\text{'}\}\{A\}\; \backslash subseteq\; X\_i$.

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

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

{{cite arXiv |last1=Bodirsky |first1=Manuel |title=Ramsey Classes: Examples and Constructions |date=27 May 2015 |class=math.CO |eprint=1502.05146}}

{{cite journal |last1=Hubička |first1=Jan |last2=Nešetřil |first2=Jaroslav | authorlink2=Jaroslav Nešetřil |title=All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms) |journal=Advances in Mathematics |date=November 2019 |volume=356 |pages=106791 |doi=10.1016/j.aim.2019.106791 |arxiv=1606.07979|s2cid=7750570 }}