A collection of families (Fthere is no pair i = j for which some F i ∈ F i is comparable to some F j ∈ F j . Two natural measures of the 'size' of such a family are the sum k i=1 |F i | and the product k i=1 |F i |. We prove new upper and lower bounds on both of these measures for general n and k ≥ 2 which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011.

Given a graph H, we say that an edge-coloured graph G is H-rainbow saturated if it does not contain a rainbow copy of H, but the addition of any non-edge in any colour creates a rainbow copy of H. The rainbow saturation number rsat(n, H) is the minimum number of edges among all H-rainbow saturated edge-coloured graphs on n vertices. We prove that for any non-empty graph H, the rainbow saturation number is linear in n, thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we also give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz.