An inside/outside Ramsey theorem and recursion theory

Inspired by Ramsey’s theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph G G contains subset H"> H encoding="application/x-tex">H such that vertex of is adjacent precisely none, one, or infinitely many vertices . We analyze the Rival–Sands from perspective reverse mathematics Weihrauch degrees. In mathematics, find equivalent arithmetical comprehension hence stronger than pairs. also identify a weak form turn degrees give finer analysis theorem’s computational strength. double jump König’s lemma. believe first natural shown exhibit exactly this Furthermore, combining our result with Brattka Rakotoniaina, obtain solving one instance corresponds simultaneously countably instances Finally, show uniform strength weaker pairs showing number well-known consequences do not reduce theorem. address apparent gap in literature concerning relationship between corresponding ascending/descending sequence principle pigeonhole principle.