Reverse mathematics and marriage problems with finitely many solutions

We analyze the logical strength of theorems on marriage problems with a fixed finite number of solutions via the techniques of reverse mathematics. We show that if a marriage problem has k solutions, then there is a finite set of boys such that the marriage problem restricted to this set has exactly k solutions, each of which extend uniquely to a solution of the original marriage problem. The strength of this assertion depends on whether or not the marriage problem has a bounding function. We also answer three questions from our previous work on marriage problems with unique solutions. Our aim is to analyze some marriage theorems via the techniques of reverse mathematics. The subsystems of second order arithmetic used are RCA0, which includes comprehension for recursive (or computable) sets of natural numbers, WKL0, which appends a weak form of König’s Lemma for trees, and ACA0, which appends comprehension for arithmetically definable sets. We refer the reader to Simpson [5] for an extensive development of the program of reverse mathematics. ∗Corresponding author: [email protected] Department of Mathematical Sciences, Appalachian State University, Boone, NC, (1)-828-262-2861, Fax: (1)-828-265-8617 †[email protected] Department of Mathematical Sciences, Appalachian State University, Boone, NC MSC: 03B30, 03F35