Un teorema de Ramsey para los enteros
Andrés Eduardo Caicedo
TL;DR
This work develops a Ramsey-theoretic analysis for linear orders, focusing on the arrow relation $L o(L_1,L_2)^2$ and its implications for infinite structures. By combining partition calculus for orders with the structure of order types (notably $oldsymbol{ extomega}$, $oldsymbol{ extomega^*}$, and their concatenations), the author isolates two fundamental suborders, $oldsymbol{ extomega}oldsymbol{ extearrow}oldsymbol{ extomega^*}$ and $oldsymbol{ extomega^*}oldsymbol{ extearrow}oldsymbol{ extomega}$, as the controlling templates for Ramsey-type results involving the integers. The main result establishes an equivalence: for any $n\ge 3$, $L o(oldsymbol{ extomega}oldsymbol{ extomega^*} ext{ or }oldsymbol{ extomega^*}oldsymbol{ extomega})^1$ if and only if $L o(oldsymbol{ ext Z},n)^2$, thereby reducing the Ramsey problem for $oldsymbol{ ext Z}$ to a finite dichotomy of suborder types. The concluding discussion surveys related work on dispersibility and total orders, highlighting open questions about the existence of universal orders with prescribed coloring properties and placing these results in the broader Ramsey-theoretic landscape.
Abstract
We find all linear orders $L,L_2$ such that $L\to(\mathbb Z,L_2)^2$. The key is the identification of the orders $L$ such that $L\to(\mathbb Z,\mathbb Z)^1$.