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

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Summary

Introduction

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

7.Conclusion and Open Conjectures
Design examples
All 6 perfect collections
Induced Balanced FHP sets
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