On Regular Set Systems Containing Regular Subsystems
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Let X,Y be finite sets, r,s,h, λ∈ℕ with s≥ r, X⊊ Y. By λXh we mean the collection of all h-subsets of X where each subset occurs λ times. A coloring of λXh is r-regular if in every color class each element of X occurs r times. A one-regular color class is a perfect matching. We are interested in the necessary and sufficient conditions under which an r-regular coloring of λXh can be embedded into an s-regular coloring of λYh. Using algebraic techniques involving glueing together orbits of a suitably chosen cyclic group, the first author and Newman (Combinatorica 38 (2018), no. 6, 1309–1335) solved the case when λ=1,r=s, (|X|,|Y|,h)=(|Y|,h). Using purely combinatorial techniques, we nearly settle the case h=4. Two major challenges include finding all the necessary conditions, and obtaining the exact bound for |Y|. It is worth noting that completing partial symmetric latin squares is closely related to the case λ =r=s=1, h=2 which was solved by Cruse (J. Comb. Theory Ser. A 16 (1974), 18–22).
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