RAINBOW PATHS AND RAINBOW MATCHINGS IN GRAPHS RON AHARONI, JOSEPH BRIGGS, JINHA KIM, AND MINKI KIM Abstract. We prove that if n ≥ 3, then any family of 3n − 3 sets of matchings of size n in any graph has a rainbow matching of size n. This improves on a previous result, in [ABC + 19], in which 3n − 3 is replaced by 3n − 2. We also prove a “cooperative” generalization: for t> 0 and n ≥ 3, any 3n − 4+ t sets of edges, the union of every t of which contains a matching of size n, have a rainbow matching of size n. 1. Introduction Given a collection of sets, S =(S 1 ,...,S m ), an S -rainbow set is the image of a partial choice function of S . So, it is a set {x i j }, where 1 ≤ i 1 <...<i k ≤ m and x i j ∈ S i j (j ≤ k). We call the sets S i colors and we say that x i j is colored by S i j and that x i j represents S i j in the rainbow set. Given numbers m, n, k, we write (m, n) → k if every m matchings of size n in any graph have a rainbow matching of size k, and (m, n) → B k if the same is true in every bipartite graph. Generalizing a result of Drisko [Dri98], the first author and E. Berger proved [AB09]: Theorem 1.1. (2n − 1,n) → B n. In [ABC + 19] it was conjectured that almost the same is true in all graphs: Conjecture 1.2. (2n, n) → n. If n is odd then (2n − 1,n) → n. If true, this is reminiscent of the relationship between K¨onig’s the- orem, stating that χ e ≤ Δ in bipartite graphs, and Vizing’s theorem, χ e ≤ Δ + 1 in general graphs, where χ e is the edge chromatic number, namely the minimal number of matchings covering the edge set of the graph. The conjecture says that there is a price of just 1 for passing from bipartite graphs to general graphs. Date : November 8, 2021. The first author was supported by BSF grant no. 2006099, an ISF grant and the Discount Bank Chair at the Technion. 1 arXiv:2004.07590v1 [math.CO] 16 Apr 2020