In the area of mathematics known as Ramsey theory, a Ramsey class[1] is one which satisfies a generalization of Ramsey's theorem.
Suppose , and are structures and is a positive integer. We denote by the set of all subobjects of which are isomorphic to . We further denote by the property that for all partitions of there exists a and an such that .
Suppose is a class of structures closed under isomorphism and substructures. We say the class has the A-Ramsey property if for ever positive integer and for every there is a such that holds. If has the -Ramsey property for all then we say is a Ramsey class.
Ramsey's theorem is equivalent to the statement that the class of all finite sets is a Ramsey class.
References
edit- ^ 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.
- ^ Bodirsky, Manuel (27 May 2015). "Ramsey Classes: Examples and Constructions". arXiv:1502.05146 [math.CO].
- ^ 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.
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