The edge covering number of the intersection of two matroids

The edge covering number of a hypergraph A is β ( A ) = min { | B | : B ⊆ A , ⋃ B = ⋃ A } . The paper studies a conjecture on the edge covering number of the intersection of two matroids. For two natural numbers k , ℓ , let f ( k , ℓ ) be the maximal value of β ( M ∩ N ) over all pairs of matroids M , N such that β ( M ) = k and β ( N ) = ℓ . In (Aharoni and Berger, 2006) [1] the first two authors proved that f ( k , ℓ ) ≤ 2 max ( k , ℓ ) and conjectured that f ( k , k ) = k + 1 and f ( k , ℓ ) = ℓ when ℓ > k . In this paper we prove that f ( k , k ) ≥ k + 1 , f ( 2 , 2 ) = 3 and f ( 2 , 3 ) ≤ 4 . We also form a conjecture on the edge covering number of 2-polymatroids that is a common extension of the above conjecture and the Goldberg–Seymour conjecture, and prove its first non-trivial case.