On the logical strengths of partial solutions to mathematical problems

TL;DR

This paper investigates the logical complexity of finding partial solutions to mathematical problems using reverse mathematics, revealing when such variants are easier or equivalent to the original problems, especially in the context of K"onig's lemma and related problems.

Contribution

It introduces the concept of Ramsey-type variants of problems and analyzes their relative difficulty, establishing equivalences and strict reductions in the framework of reverse mathematics.

Findings

Ramsey-type variants can be strictly easier or equivalent to original problems.

Ramsey-type weak K"onig's lemma is robust and equivalent to several perturbations.

Ramsey-type weak weak K"onig's lemma relates to diagonally non-recursive functions and is easier than the original.

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.