The Intersection of Two Infinite Matroids
Ron Aharoni, Ziv Ran · Journal of the London Mathematical Society · 1998
Conjecture: Let M and N be two matroids (possibly of infinite ranks) on the same set S. Then there exists a set I independent in both M and N, which can be partitioned as I=H∪K, where spM(H)∪spN(K)=S. This conjecture is an extension of Edmonds' matroid intersection theorem to the infinite case. We prove the conjecture when one of the matroids (say M) is the sum of countably many matroids of finite rank (the other matroid being general). For the proof we have also to answer the following question: when does there exist a subset of S which is spanning for M and independent in N?