Rainbow forest packing at the boundary k = 2m

Determine whether, for every ε>0, there exists m₀ such that every properly edge-colored simple graph G with e(G)=km and maximum color-class size μ(G,c)≤m, where m≥m₀ and 1≤k≤2m, contains at least (1−ε)m pairwise edge-disjoint rainbow forests, each with exactly k edges.

Background

The paper proves that if 0<δ<1 is fixed, m is sufficiently large, and 1≤k≤(2−δ)m, then every properly edge-colored simple graph with km edges and color classes of size at most m contains at least (1−ε)m pairwise edge-disjoint rainbow forests with exactly k edges. The factor m is asymptotically optimal because the graph has only km edges.

The authors show that the endpoint k=2m+1 cannot be included: a copy of K_{2m+1} with every edge assigned a distinct color has m(2m+1)=km edges and satisfies the global color bound, but no forest in this graph has 2m+1 edges. The unresolved boundary is therefore whether the theorem remains valid all the way through k=2m, removing the fixed relative margin δ while stopping short of the known obstruction.