On the logical strengths of partial solutions to mathematical problems
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 Ramseytype 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 Ramseytype variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramseytype variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramseytype variant of a problem is equivalent to the original problem. We show that the Ramseytype variant of weak König's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramseytype weak König's lemma and algorithmic randomness by showing that Ramseytype weak weak König's lemma is equivalent to the problem of finding diagonally nonrecursive functions and that these problems are strictly easier than Ramseytype weak König's lemma. This answers a question of Flood.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 arXiv:
 arXiv:1411.5874
 Bibcode:
 2014arXiv1411.5874B
 Keywords:

 Mathematics  Logic;
 03B30;
 03F35
 EPrint:
 43 pages