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Hyperedge intersection consequence in Construction 1

PDF p. 2: Construction 1, final sentence

Statement

Let q≥3 be an integer and let H be a hypergraph satisfying the conditions of Construction 1, with Berge girth at least q as specified in the hypergraph terminology. Then any two distinct hyperedges of H intersect in at most one vertex.

Facts & Assumptions

[F1]

By Hypergraph terminology used in Construction 1, the hypergraph in this formulation is finite, and its girth is Berge girth, with infinite girth permitted when there is no Berge cycle.

[F2]

By Berge girth at least three forces linearity, a finite hypergraph of Berge girth at least three is linear: any two distinct hyperedges have at most one common vertex.

Proof

Given: An integer q≥3 and a hypergraph H satisfying the stated construction conditions, including Berge girth at least q.

Goal: Any two distinct hyperedges of H intersect in at most one vertex.

1.1givenF1

The hypergraph H is finite and has Berge girth at least three, since its Berge girth is at least q and q≥3. This also holds if its girth is infinite.

2.1F2step 1.1∎

Applying the linearity lemma to H shows that any two distinct hyperedges have at most one common vertex, as required.

Depends on

Dependency tree · 2 levels

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Sources