Girth Conditions and Rota’s Basis Conjecture

Rota’s basis conjecture (RBC) states that given a collection ℬ of n bases in a matroid M of rank n , one can always find n disjoint rainbow bases with respect to ℬ . In this paper, we show that if M has girth at least n-o(√(n)) , and no element of M belongs to more than o(√(n)) bases in ℬ , then one can find at least n - o(n) disjoint rainbow bases with respect to ℬ . More specifically, we show that if M has girth at least n- β (n) +1 and each element belongs to no more than κ (n) bases in ℬ , then letting γ (n) = 4(κ (n) + β (n)+1)^2, one can find at least n - γ (n) disjoint rainbow bases provided 2γ (n) < n . This result can be seen as an extension of the work of Geelen and Humphries, who proved RBC in the case where M is paving, and ℬ is a pairwise disjoint collection. The proofs here are based on modifications to the cascade idea introduced by Bucić et al.