# Aharoni, Ron

## Strong matchings and covers ★★★

Author(s): Aharoni

Let be a hypergraph. A strongly maximal matching is a matching so that for every matching . A strongly minimal cover is a (vertex) cover so that for every cover .

Conjecture   If is a (possibly infinite) hypergraph in which all edges have size for some integer , then has a strongly maximal matching and a strongly minimal cover.

Keywords: cover; infinite graph; matching

## Aharoni-Berger conjecture ★★★

Author(s): Aharoni; Berger

Conjecture   If are matroids on and for every partition of , then there exists with which is independent in every .

Keywords: independent set; matroid; partition

## Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.

Conjecture   If is the maximal degree of a graph , then is strongly -colorable.

Keywords: strong coloring