Symmetry and the Ramsey degree of posets
If \(P\) is a finite poset, let \(e(P)\) be the number of linear extensions of \(P\) and \(\Aut(P)\) the group of (order) isomorphisms of \(P\). The author defines \(t(P)= e(P)/|\Aut(P)|\) and he also shows that \(t(P)\) is an integer, the Ramsey degree of \(P\) and he observes that if \(t(P)=1\), then for each poset \(Q\) and each natural number \(r\), there is a poset \(R\) such that for every \(r\)-coloring of the set of copies of \(R\) in \(P\), there is a monochromatic copy of \(Q\) in \(R\) (per J. Nešetřil and V. Rödl). Using this interesting definition and the approach leading to the above characterization his main theorem is Theorem 1: For posets \(P\) and \(Q\) and a natural number \(r\), there exists a poset \(R\) such that for any \(r\)-coloring \(\chi\) of the copies of \(P\) in \(R\), there exists a copy \(Q'\) of \(Q\) in \(R\) such that \(\chi\) assumes at most \(t(P)\) values on the set of copies of \(Q'\) in \(P\). A copy of a poset \(A\) in a poset \(B\) is the image of \(A\) in \(B\) via an order-preserving injection.
scientific article; zbMATH DE number 1018522