Rainbow Sets in the Intersection of Two Matroids

Given sets F 1 , … , F n , a partial rainbow set is the range of a partial choice function, where if the same element x is chosen from k different F i ʼs it is considered as repeating k times. Aharoni and Berger [R. Aharoni and E. Berger, unpublished] conjectured that if M and N are matroids on the same ground set, and F 1 , … , F n are sets of size n belonging to M ∩ N , then there exists a rainbow set of size n − 1 belonging to M ∩ N . Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least n − n .