On regular set systems containing regular subsystems

Let X, Y be finite sets, r, s, h, lambda is an element of N with s >= r, X subset of Y. By lambda(X ) h we mean the collection of all h-subsets of X where each subset occurs lambda times. A coloring (partition) of lambda(X ) is r-regular if each h element of X is in exactly r subsets of each color. A one-regular color class is a perfect matching. We are interested in necessary lambda(X and sufficient conditions under which an r-regular coloring of ) can be embedded into an s-regular coloring of lambda(Y ). Using h h algebraic techniques involving glueing together orbits of a suitably chosen cyclic group, the first author and Newman solved the case when lambda = 1, r = s, gcd(|X|, |Y|, h) = gcd(|Y|, h). Using purely combinatorial techniques, we nearly settle the case h = 4. It is worth noting that completing partial symmetric latin squares is closely related to the case lambda = r = s = 1, h = 2 which was solved by Cruse. (c) 2021 Elsevier Ltd. All rights reserved.