Ramsey’s theorem on countable infinite sets states that for all natural numbers n, for all finite colorings of the n-element subsets of some infinite countable set, there exists an infinite countable homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let S be a linearly ordered set and \(n\in \mathbb {N}\) . The big Ramsey degree of \(n\) in S, denoted \(T(n,S)\) , is the least natural number t such that, for any finite coloring of the size \(n\) subsets of S, there exists \(S'\subseteq S\) such that (i) \(S'\) is order-equivalent to S, and (ii) if the coloring is restricted to the size \(n\) subsets of \(S'\) then at most t colors are used. Mašulović & Šobot (2021) showed that \(T(n,\omega +\omega )=2^{n}\) . From this one can obtain \(T(n,\zeta )=2^{n},\) where \(\zeta \) is the ordered set of integers. We give a direct proof that \(T(n,\zeta )=2^{n}\) . Mašulović and Šobot (2021) also showed that for all countable ordinals \(\alpha < \omega ^\omega \) and \(n\in \mathbb {N}\) , \(T(n,\alpha )\) is finite. We find exact values of \(T(n,\alpha )\) for all ordinals \(\alpha \) less than \(\omega ^\omega \) and all \(n\in \mathbb {N}\) .
