For rank-n matroids with girth at least n-o(√n) and no element in more than o(√n) bases, the paper claims there are n-o(n) disjoint rainbow bases.
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Girth conditions and Rota's basis conjecture
For rank-n matroids with girth at least n-o(√n) and no element in more than o(√n) bases, the paper claims there are n-o(n) disjoint rainbow bases.