Abstract
Enumeration of 5-Uniform Hypergraph Designs with Edge 6-Distant Condition
Highlights
Problem 1.1 Let n ≥ 3 be integer, set S = [n], T = {n + 1, n + 2, · · ·, n + λ}.For integer λ, there exists a 5−uniform (3, 2)−bipartite hypergraph H = (S ∪ T, E), which satisfies that: 1. |E| = n 3; 2. ∀e ∈ E, |e ∩ S| = 3, |e ∩ T | = 23. ∀e1, e2 ∈ E, |e1 ∩ e2| ≤ 2For a given integer n, determine the minimum of λ that H :exists
Before we prove the uniqueness of these designs, we’ll introduce some behavior of balanced FHP sets. and we’ll observe how FHP’s triples are distributed in induced 6 balanced FHP sets in Appendix C
These works are on some small scales of n, it is shown that there are some kinds of patterns in the designs
Summary
We’ll discover the connections among the perfect collections, balanced FHP sets, and 6-distant designs, which is needed in the prove of the uniqueness. Corollary 5.18 For a 6-distant design H = (V, E), Let F(H) = {Fi = {e ∩ S|i ∈ e ∈ E}|i ∈ T } be the balanced FHP set induced by H. Lemma 5.20 For a perfect collection P, FP = {FP , FP |P ∈ P} is a balanced FHP set. Proof: For a perfect collection P, by Lemma (5.20), FP is a balanced FHP set. We’ll observe how FHP’s triples are distributed in induced 6 balanced FHP sets in Appendix C These properties are necessary for proving the uniqueness. For each FHP F, F is in 2 of the 6 balanced FHP sets induced by these 6 perfect collections
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