# Aharoni-Berger conjecture

Is it true that if [math]M_1, \ldots, M_k[/math] are matroids on the same ground set *S* and [math]\sum_{i=1}^k r_i (X_i) \geq (k-1)l[/math] for all partitions [math]X_1, \ldots, X_k[/math] of *S*, then there exists a common independent set of size [math]l[/math]?

## Remarks

Ryser's conjecture asks whether for any *k*-uniform *k*-partite hypergraph [math]\tau \leq (k-1)\nu[/math] holds, where [math]\tau[/math] is the minimum size of a vertex cover and [math]\nu[/math] is the maximum number of disjoint hyperedges. The case *k*=2 is just Kőnig's theorem, while the case *k*=3 was proved by Aharoni ^{[1]}. Just as Edmonds' matroid intersection theorem generalizes Kőnig's theorem, the Ryser conjecture is generalized by the Aharoni-Berger conjecture. It was proved for *k*=3 by Aharoni and Berger ^{[2]}.

## References

- ↑ R. Aharoni,
*Ryser's conjecture for tripartite 3-graphs*, Combinatorica 21 (2001), 1-4. DOI link - ↑ R. Aharoni, E. Berger,
*The intersection of a matroid with a simplicial complex*, Trans. Amer. Math. Soc. 358 (2006), 4895-4917. DOI link, JSTOR link