Indicators, Chains, Antichains, Ramsey Property

Abstract

We introduce two Ramsey classes of finite relational structures. The first class contains finite structures of the form $\left( A,\,\left( {{I}_{i}} \right)_{i=1}^{n},\le ,\left( {{\underline{\prec }}_{i}} \right)_{i=1}^{n} \right)$, where $\le$ is a total ordering on $A$ and ${{\underline{\prec }}_{i}}$ is a linear ordering on the set $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$. The second class contains structures of the form a $\left( a,\le ,\left( {{i}_{i}} \right)_{i=1}^{n},\underline{\prec } \right)$, where $\left( A,\,\le \right)$ is a weak ordering and $\underline{\prec }$ is a linear ordering on $A$ such that $A$ is partitioned by $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ into maximal chains in the partial ordering $\le$ and each $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ is an interval with respect to $\underline{\prec }$.