Abstract
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to K\"onig's lemma, such as restrictions of K\"onig's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak K\"onig's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak K\"onig's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak K\"onig's lemma and algorithmic randomness by showing that Ramsey-type weak weak K\"onig's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak K\"onig's lemma. This answers a question of Flood.
Highlights
This work presents a detailed study of the question given some mathematical problem, is it easier to find an infinite partial solution than it is to find a complete solution? that was implicitly raised by Flood’s work in [9]
We answer Flood’s question by showing that RWKL is strictly stronger than diagonally non-recursive (DNR) (Corollary 6.12 below), and we show that DNR is equivalent to the Ramsey-type variant of weak weak König’s lemma
We propose that RWKL is fundamental, in no small part because the basic question that inspires RWKL, that is, the question of whether or not it is easier to find an infinite partial solution to a problem than to find a complete solution, is so natural
Summary
This work presents a detailed study of the question given some mathematical problem, is it easier to find an infinite partial solution than it is to find a complete solution? that was implicitly raised by Flood’s work in [9]. For example, Ramsey-type König’s lemma is the problem of producing an infinite partial path (in the sense described above) through an infinite, finitely-branching tree. We show that the forgoing example of König’s lemma for arbitrary infinite, finitely branching trees is equivalent to its Ramsey-type variant (Theorem 3.17 below). We answer Flood’s question by showing that RWKL is strictly stronger than DNR (Corollary 6.12 below), and we show that DNR is equivalent to the Ramsey-type variant of weak weak König’s lemma (which is König’s lemma restricted to binary branching trees of positive measure; Theorem 3.4 below).. In these sections, we consider statements that are equivalent to weak König’s lemma (compactness for propositional logic in Section 4 and graph coloring in Section 5) and show that their corresponding Ramsey-type variants are equivalent to RWKL.
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