Path or Fail? Generalizing Menger’s Theorem to Distance k Stable Sets on Paths

Abstract

This paper is a survey of results generalizing Menger’s Theorem to AB-paths of pairwise distance k apart. First, we describe several methods of proving Menger’s theorem. Then, we discuss upper bounds on Menger-like requirements for paths of pairwise distance k, including researched results for k = 2, n = 2, and n = k = 3. New proofs from my advisor Dr. Maria Chudnovsky are presented for the k = 2 and k = 3 cases.