Open Problem Garden

Importance: High ✭✭✭

Author(s):	Aharoni, Ron
	Berger, Eli

Subject:	Combinatorics
	» Matroid Theory

Keywords:	independent set
	matroid
	partition

Recomm. for undergrads: no

Posted	by:	mdevos
	on:	June 11th, 2007

Conjecture If $M_1,\ldots,M_k$ are matroids on $E$ and $\sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1)$ for every partition $\{X_1,\ldots,X_k\}$ of $E$ , then there exists $X \subseteq E$ with $|X| = \ell$ which is independent in every $M_i$ .

Let us begin by stating two classic results. For a graph (or hypergraph) we let $\tau$ denote the size of the smallest (vertex) cover and we let $\nu$ denote the size of the largest matching.

Theorem (König) $\nu = \tau$ for every bipartite graph.

Theorem (Matroid Intersection) If $M_1,M_2$ are matroids on $E$ and $rk_{M_1}(X_1) + rk_{M_2}(X_2) \ge \ell$ for every partition $\{X_1,X_2\}$ of $E$ , then there exists $X \subseteq E$ with $|X| = \ell$ which is independent in both $M_1$ and $M_2$ .

The matroid intersection theorem is exactly the $k=2$ case of the above conjecture, but it may also be viewed as a generalization of König's theorem. To see this, let $G$ be a bipartite graph with edge set $E$ and bipartition $\{A_1,A_2\}$ and for $i=1,2$ let $M_i$ be the (uniform) matroid on $E$ where a subset $S \subseteq E$ is independent if no two edges in $S$ share an endpoint in $A_i$ . Then $rk_{M_i}(S)$ is the number of vertices in $A_i$ which are incident with an edge in $S$ , so $rk_{M_1}(X_1) + rk_{M_2}(X_2)$ has minimum value $\tau$ , and a set of edges is independent in both $M_1$ and $M_2$ if and only if it is a matching, so the size of the largest such set is $\nu$ .

A famous conjecture of Ryser suggests a generalization of König's theorem to hypergraphs. It claims that every $k$ -partite $k$ -uniform hypergraph satisfies $\tau \le (k-1) \nu$ . The above conjecture is the common generalization of this conjecture of Ryser and the matroid intersection theorem. Aharoni [A] proved the 3-partite 3-uniform case of Ryser's conjecture, and this was extended by Aharoni-Berger [AB] to the $k=3$ case of the above conjecture. The conjecture remains open for $k \ge 4$ .