核心概念
By leveraging constructions from coding theory, we obtain improved lower bounds on the maximum number of edges in r-uniform hypergraphs with girth 5 and 6, for all r ≥ 3.
要約
The paper studies the maximum number of edges in r-uniform hypergraphs with girth 5 and 6, denoted by exr(N, C<5) and exr(N, C<6) respectively, where N is the number of vertices and C<g denotes the family of Berge cycles of length at most g-1.
Key highlights:
- The authors address an unproved claim from prior work that the lower bound exr(N, C<5) = Ωr(N^{3/2-o(1)}) holds for all r ≥ 3. They identify an obstacle in the claimed proof and show that this obstacle can be overcome when r ∈ {4, 5, 6}.
- For all other r, the authors use constructions from coding theory to prove new lower bounds on exr(N, C<5) and exr(N, C<6) that improve upon the previous probabilistic bounds.
- The authors also show that recent results on hypergraph Turán problems can be used to improve the sphere packing bound for linear codes of distance 6.
The paper provides a comprehensive analysis of the connections between hypergraph Turán problems and coding theory, leading to new insights and improved bounds in both areas.