For any graph G, the minimum universe size for G-Sperner families equals the chromatic number of G, while G-cover-free families on paths and cycles satisfy log2(n) ≤ t ≤ 1.893 log2(n) + O(1).
On the upper bound of the size of the r-cover-free families.Journal of Combinatorial Theory, Series A, 66(2):302–310, 1994
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Cover-free families on graphs
For any graph G, the minimum universe size for G-Sperner families equals the chromatic number of G, while G-cover-free families on paths and cycles satisfy log2(n) ≤ t ≤ 1.893 log2(n) + O(1).